As prosecutors in Manhattan weigh whether to charge Donald J. Trump with fraud, they have zeroed in on financial documents that he used to obtain loans and boast about his wealth, according to people with knowledge of the matter.

The documents, compiled by Mr. Trump’s longtime accountants and known as annual statements of financial condition, could help answer a question at the heart of the long-running criminal investigation into the former president: Did he inflate the value of his assets to defraud his lenders?

In recent weeks, prosecutors in the office of the Manhattan district attorney, Cyrus R. Vance Jr., have questioned one of Mr. Trump’s accountants before a grand jury as part of their examination of the financial statements, said the people with knowledge of the matter. Prosecutors also interviewed his longtime banker, another person said.

If the prosecutors seek an indictment, the case’s outcome could hinge on whether they can use the documents to prove that a defining feature of Mr. Trump’s public persona — his penchant for hyperbole — was so extreme and intentional when dealing with his lenders that it crossed the line into fraud.

Whenever Mr. Trump needed a loan, he would provide potential lenders with the statements, which contained optimistic projections about the value of his real estate business as well as sweeping disclaimers noting the numbers’ limitations.

Mr. Vance’s prosecutors found that the accountants who put together the statements relied on underlying information provided by the Trump Organization, Mr. Trump’s family business, according to the people with knowledge of the matter, who were familiar with the questions prosecutors asked and spoke only on condition of anonymity because they were discussing confidential testimony.

The prosecutors, working with the office of the New York State attorney general, Letitia James, have examined the possibility that Mr. Trump and his deputies at the company cherry-picked favorable information — and ignored data that ran counter to it — to essentially mislead the accountants into presenting an overly rosy picture of his finances.

While the numbers could implicate Mr. Trump, disclaimers in the statements that the data had not been audited or authenticated could help his defense, underscoring the challenge that prosecutors face as they grapple with whether to charge the former president.

A spokeswoman for Mr. Trump’s accounting firm, Mazars USA, declined to comment beyond saying that it could not discuss its clients or its work for them without their consent, and that Mazars remained “committed to fulfilling all of our professional and legal obligations.”

Jerry D. Bernstein, a lawyer for Mazars, which has assembled Mr. Trump’s personal and corporate tax returns for years, declined to elaborate.

A spokesman for the Manhattan district attorney’s office declined to comment, as did a lawyer for Mr. Trump, Ronald P. Fischetti. A spokeswoman for the Trump Organization said New York was struggling with crime and homelessness “yet the only focus on the New York D.A. is to investigate Trump for political gain.”

Mr. Trump did not personally assemble the data for the accountants, but the documents left no doubt as to who was accountable for their contents: “Donald J. Trump is responsible for the preparation and fair presentation of the financial statement in accordance with accounting principles generally accepted in the United States of America,” his accountants wrote in a cover letter attached to the statements in 2011 and 2012.

Yet the accountants also acknowledged they “have not audited or reviewed” the information and “do not express an opinion or provide any assurance about” it, a common caveat in statements of financial condition. The accountants disclosed that, while compiling the information for Mr. Trump, they had “become aware of departures from accounting principles generally accepted in the United States of America.”

Armed with those caveats, Mr. Trump’s lawyers would most likely argue that no one, let alone sophisticated lenders, should have taken his valuations at face value. And even if his valuations were false, the lawyers might argue, the lenders conducted their own analyses of Mr. Trump’s assets and concluded that he was a worthy borrower.

Mr. Trump’s lawyers could also call on people with expertise in property assessments to say that the value of a hotel or office building may be subject to various interpretations.

The public got a glimpse of Mr. Trump’s statements when his former lawyer and fixer, Michael D. Cohen, released them when he testified to Congress in 2019.

Mr. Cohen, who split from Mr. Trump during the presidency and eventually pleaded guilty to several federal crimes, told Congress that “Mr. Trump inflated his total assets when it served his purposes such as trying to be listed amongst the wealthiest people in Forbes and deflated his assets to reduce his real estate taxes.”

Mr. Cohen provided Congress with Mr. Trump’s statements from 2011 to 2013. Mr. Trump, he said, had provided the documents to Deutsche Bank when inquiring about a potential loan to buy the Buffalo Bills.

The deal never materialized, but Mr. Vance’s prosecutors have questioned witnesses about Mr. Trump’s statements to Deutsche Bank during the process, the people said. They have questioned Mr. Cohen and an employee from Deutsche Bank, Mr. Trump’s main lender.

For years, Mr. Trump shared the statements with Deutsche Bank and other potential lenders to offer a glowing assessment of his financial health when he needed a loan for a hotel, golf course or office building.

But before impressing his lenders, Mr. Trump had to have his employees assemble spreadsheets detailing the underlying value of his assets, according to people with knowledge of the process. The employees would then send the spreadsheets to his accounting firm, Mazars, which would compile the information into the annual statements.

The statements, issued as of June 30 every year, often began with a one-page list of Mr. Trump’s assets. Each property — Trump Tower, his golf clubs, his hotels — was listed next to a dollar amount that represented its supposed value. His cash and investments received a value, too, as did the Miss Universe pageants and other assets.

The second and final page of the 2011 and 2012 statements described Mr. Trump’s liabilities — essentially a list of any outstanding loans — and then reported his net worth. In 2011, Mr. Trump claimed a net worth of $4.2 billion. In 2012, it was more than $4.5 billion.

In determining the value of a property owned by Mr. Trump, his employees often looked at recent selling prices of comparable buildings, a common real estate valuation method.

But prosecutors have questioned whether the Trump Organization routinely selected the most valuable properties, even if they were not completely comparable, and disregarded sales of buildings that would have dragged down Mr. Trump’s valuations, the people with knowledge of the matter said.

The prosecutors have also scrutinized how the company projected future income that was not guaranteed, the people said.

Some of the irregularities that appeared in the statements were relatively trivial — Mr. Trump claimed, as he often has, that Trump Tower was 68 stories tall when it was really 58 — while others raised larger questions about the legitimacy of the numbers. The 2011 statement omitted his hotel in Chicago — and the tens of millions of dollars in debt Mr. Trump had personally guaranteed on the property.

The 2012 statement’s cover letter detailed a long list of disclaimers, including that the statement contained predictions about the future and that it did not include data for the Chicago hotel. The omission, the accountants suggested, ran contrary to an important rule of thumb: “Accounting principles generally accepted in the United States of America require that personal financial statements include all assets and liabilities.”

The accountants concluded the letter with a broad note of caution: “Users of this financial statement should recognize that they might reach different conclusions about the financial condition of Donald J. Trump.”

In addition to the Mazars cover letter, Mr. Trump added his own addendum to the statements of financial condition, a series of explanations and caveats that referred to the values in the statements as estimates.

His lawyers are likely to contend that these caveats absolve Mr. Trump of criminal liability. They could also argue that the lenders are not actually victims, highlighting the fact that his main lender, Deutsche Bank, made money in its dealings with Mr. Trump. (The Trump Organization is expected to soon sell the lease on its hotel in Washington for at least $375 million and pay off Deutsche Bank’s loan to that property.)

But even the former president’s own notes point to his involvement in determining the values that would be represented.

The value of Trump Tower, the notes said, was based partly on “an evaluation by Mr. Trump.”

David Enrich contributed reporting.

A long-running criminal investigation into Donald J. Trump and his family business is reaching a critical phase as Cyrus R. Vance Jr., the prosecutor overseeing the inquiry, enters his final weeks as Manhattan district attorney.

Mr. Vance’s prosecutors have issued new subpoenas for records about Mr. Trump’s hotels, golf clubs and office buildings. They recently interviewed a banker employed by Deutsche Bank, Mr. Trump’s top lender. And earlier this month, they told a top Trump executive who had been under scrutiny, Matthew Calamari, that they did not currently plan to indict him in the purported tax-evasion scheme that led to charges against Mr. Trump’s company and its chief financial officer.

The developments, described by people with knowledge of the matter, show that the Manhattan prosecutors have shifted away from investigating those tax issues and returned to an original focus of their three-year investigation: Mr. Trump’s statements about the value of his assets.

In particular, the people said, the prosecutors are zeroing in on whether Mr. Trump or his company inflated the value of some of his properties while trying to secure financing from potential lenders. If Mr. Vance’s office concludes that Mr. Trump intentionally submitted false values to potential lenders, prosecutors could argue that he engaged in a pattern of fraud. 

Mr. Trump and his company have denied wrongdoing, calling the inquiry a politically motivated witch hunt by Mr. Vance, a Democrat. His prosecutors are working with the office of the New York State attorney general, Letitia James, also a Democrat, who is running for governor and has been a vocal critic of Mr. Trump.

The investigation is playing out as Mr. Trump continues to wield enormous influence over the Republican Party and flirts with another presidential run in 2024. It has yet to hamper his political standing and could even energize his base, though the inquiry could also serve as a costly distraction.

It is unclear whether any new charges will be brought, but if they are, it could be difficult to prove that Mr. Trump or his company defrauded their lenders. One challenge is that the lenders are sophisticated financial institutions that most likely conducted their own assessments of Mr. Trump’s property values without relying entirely on him.

Still, Mr. Vance’s prosecutors have issued a flurry of subpoenas in recent months.

At least one subpoena issued this summer to Mr. Trump’s company, the Trump Organization, demanded information about how the company valued various assets, according to the people, who all spoke on the condition of anonymity because they were not authorized to discuss the matter.

The company is resisting turning over some documents, a matter that is the subject of sealed litigation in Manhattan. 

The Deutsche Bank employee, a banker in the division that works with Mr. Trump, also recently met with prosecutors in Mr. Vance’s office to answer their questions, the people said.

The bank has issued hundreds of millions of dollars in loans to Mr. Trump over the years, including for his hotels in Chicago and Washington and his Doral golf resort in Florida.

Last year, during a relatively early phase in the investigation, the prosecutors questioned Deutsche Bank employees about the lender’s general procedures for making loan decisions.

This summer, the prosecutors diverged from that focus on valuations when they indicted the Trump Organization and its long-serving chief financial officer, Allen H. Weisselberg, for unrelated tax crimes involving a scheme to pay off-the-books luxury perks to certain executives.

Mr. Vance’s office had also been weighing for months whether to indict Mr. Calamari, Mr. Trump’s former bodyguard who now serves as the organization’s chief operating officer, for failing to pay taxes on the same sort of perks.

But this month, after prosecutors pivoted back to the broader investigation, they told Mr. Calamari that they had “no present intention” to bring charges, his lawyer said. 

“We believe that is the fair and just decision,” said the lawyer, Nicholas Gravante Jr., who had argued that Mr. Calamari was not involved in the company’s finances and relied on another Trump employee to prepare his taxes. “What the future holds in terms of the district attorney’s continuing investigation is anyone’s guess.”

Mr. Vance, who did not seek re-election for a fourth term this year, originally signaled that he wanted to decide whether to charge Mr. Trump before departing.

But he is running out of time.

He leaves office at the end of this year, and it might take prosecutors months to present a complex financial case like this one to a grand jury should they decide charges are warranted.

And so, after more than three years of investigating Mr. Trump in fits and starts, Mr. Vance may have to hand over the culmination of the case to his successor, Alvin Bragg.

In his first weeks in office, Mr. Bragg would face a monumental decision: whether to seek an indictment against a former American president.

Mr. Bragg, a former federal prosecutor who will be the first Black person to hold the office, campaigned in part on his aggressive record toward Mr. Trump, frequently mentioning the many times he had sued the Trump administration while serving in the state attorney general’s office.

Since winning the Democratic primary in June, Mr. Bragg has been more reserved. A spokesman for Mr. Bragg, Richie Fife, said that he intended to retain the team working on the Trump investigation, including two senior officials overseeing it, Mark F. Pomerantz and Carey Dunne, Mr. Vance’s general counsel who has been deeply involved in the case. 

Mr. Bragg has not yet been briefed on the details of the investigation, Mr. Fife said. He takes office on Jan. 1.

Mr. Vance is one of three district attorneys investigating Mr. Trump or his company. Mr. Trump’s attempts to overturn Georgia’s election results last year are the subject of a criminal investigation in Atlanta. And the district attorney in suburban Westchester County, outside of New York City, has begun to examine financial dealings at a golf club Mr. Trump owns there.

The Westchester investigation has focused partly on whether the Trump Organization, while seeking to cut that club’s tax bill, falsely undervalued the property. The company has denied all wrongdoing and other golf course operators say that it is common in the industry to propose lowball figures when appealing property taxes.

At one point, Mr. Vance’s office similarly examined the company’s dealings with property tax authorities in New York City, but his prosecutors appear to have largely moved away from that effort.

They are instead focused now on the company’s statements to lenders, and have also examined some of its unsuccessful attempts to secure financing, the people said.

Lawyers for the Trump Organization likely will argue that property values are subjective and that Mr. Trump could not have duped Deutsche Bank because the lender, knowing Mr. Trump’s values were estimates and that he had a reputation for exaggerating his wealth, did its own analysis of his net worth.

People with knowledge of the relationship have said that after the bank came to believe that Mr. Trump was overvaluing some of his assets, it still decided to lend him money, concluding that he was a safe risk in part because he had more than enough money and other assets to personally guarantee the debt.

The Trump Organization is expected to soon sell the lease on the hotel in Washington and pay off Deutsche Bank’s loan to that property.

Mr. Vance’s investigation began in August 2018 with a focus on the company’s role in paying hush money to an adult film actress who said she had an affair with Mr. Trump.

The inquiry was first delayed for nearly a year at the request of federal authorities, who had been focused on some similar conduct, and then by an 18-month battle in which Mr. Trump sought to block the district attorney from obtaining his tax returns.

That bitter legal battle twice reached the United States Supreme Court before ending in victory for Mr. Vance’s office, which obtained millions of pages of Mr. Trump’s personal and corporate tax records and other financial documents.

Once the prosecutors obtained the records, they focused on pressuring Mr. Weisselberg to cooperate and turn on his longtime employer.

But Mr. Weisselberg rebuffed the prosecutors, a decision that led to his indictment in July, along with the Trump Organization.

By bringing the charges against Mr. Weisselberg and the company, the prosecutors chose to take what amounted to a detour, temporarily directing attention away from the wider investigation of Mr. Trump and his family business.

That case focused on what prosecutors described as a 15-year scheme at the Trump Organization to pay its executives off-the-books perks like free cars and apartments.

Prosecutors accused Mr. Weisselberg of receiving a rent-free apartment, private school tuition for his grandchildren and leased Mercedes-Benz vehicles for him and his wife.

He failed to pay taxes on $1.7 million of those benefits, according to the indictment, which charged him with grand larceny, tax fraud and other charges.

Mr. Weisselberg’s lawyers, Mary E. Mulligan and Bryan C. Skarlatos, have said that Mr. Weisselberg will fight the charges in court.

As President Trump delivered his inaugural address in 2017, a slight woman with feathered gray hair sat listening, bundled in a hooded white parka in a fenced-off V.I.P. section. Her name was Rosemary T. Vrablic. She was a managing director at Deutsche Bank and one of the reasons Mr. Trump had just taken the oath of office.

It was a moment of celebration — and a moment of worry for Ms. Vrablic’s employer.

Mr. Trump and Deutsche Bank were deeply entwined, their symbiotic bond born of necessity and ambition on both sides: a real estate mogul made toxic by polarizing rhetoric and a pattern of defaults, and a bank with intractable financial problems and a history of misconduct.

The relationship had paid off. Mr. Trump used loans from Deutsche Bank to finance skyscrapers and other high-end properties, and repeatedly cited his relationship with the bank to deflect political attacks on his business acumen. Deutsche Bank used Mr. Trump’s projects to build its investment-banking business, reaped fees from the assets he put in its custody and leveraged his celebrity to lure clients.

Then Mr. Trump won the 2016 election, and the German bank shifted into damage-control mode, bracing for an onslaught of public scrutiny, according to several people involved in the internal response.

In the weeks before Ms. Vrablic attended his swearing-in, the bank commissioned reports to figure out how it had gotten in so deep with Mr. Trump. It issued an unusual edict to its Wall Street employees: Do not publicly utter the word “Trump.”

More than two years later, Mr. Trump’s financial ties with Deutsche Bank are the subject of investigations by two congressional committees and the New York attorney general. Investigators hope to use Deutsche Bank as a window into Mr. Trump’s personal and business finances.

Deutsche Bank officials have quietly argued to regulators, lawmakers and journalists that Mr. Trump was not a priority for the bank or its senior leaders and that the lending was the work of a single, obscure division. But interviews with more than 20 current and former Deutsche Bank executives and board members, most of them with direct knowledge of the Trump relationship, contradict the bank’s narrative.

Over nearly two decades, Deutsche Bank’s leaders repeatedly saw red flags surrounding Mr. Trump. There was a disastrous bond sale, a promised loan that relied on a banker’s forged signature, wild exaggerations of Mr. Trump’s wealth, even a claim of an act of God.

But Deutsche Bank had a ravenous appetite for risk and limited concern about its clients’ reputations. Time after time, with the support of two different chief executives, the bank handed money — a total of well over $2 billion — to a man whom nearly all other banks had deemed untouchable.

Kerrie McHugh, a Deutsche Bank spokeswoman, said: “We remain committed to cooperating with authorized investigations.”

The White House referred questions to the Trump Organization. A company spokeswoman, Amanda Miller, declined to comment.

‘A Great Son!’

In the late 1990s, Deutsche Bank, which is based in Germany, was trying to make a name for itself on Wall Street. Its investment-banking division went on a hiring binge.

The bank recruited a handful of Goldman Sachs traders to lead a push into commercial real estate. One was Justin Kennedy, the son of Supreme Court Justice Anthony Kennedy. Another was Mike Offit, whose father was the writer Sidney Offit.

At Deutsche Bank, Mr. Offit’s mandate was to lend money to big real estate developers, package the loans into securities and sell the resulting bonds to investors. He said in an interview that one way to stand out in a crowded market was to make loans that his rivals considered too risky.

In 1998, a broker contacted him to see if he would consider lending to a Wall Street pariah: Mr. Trump, who was then a casino magnate whose bankruptcies had cost banks hundreds of millions of dollars.

Mr. Offit took the meeting.

A few days later, Mr. Offit’s secretary called him. “Donald Trump is in the conference room,” she whispered. Mr. Offit said he rushed in, expecting to find an entourage. Mr. Trump was alone.

He was looking for a $125 million loan to pay for gut renovations of 40 Wall Street, his Art Deco tower in Lower Manhattan. Mr. Offit was impressed by the pitch, and the loan sailed through Deutsche Bank’s approval process.

Mr. Trump seemed giddy with gratitude, Mr. Offit recalled. He took Mr. Offit golfing. He flew him by helicopter to Atlantic City for boxing matches. He wrote a grateful note to Sidney Offit for having “a great son!”

Mr. Offit commissioned a detailed model of 40 Wall Street. A golden plaque on its pedestal bore the names and logos of Deutsche Bank and the Trump Organization. Mr. Offit gave one to Mr. Trump and kept another in his office.

Mr. Trump soon came looking for $300 million for the construction of a skyscraper across from the United Nations headquarters. The loan was approved. He wanted hundreds of millions more for his Trump Marina casino in Atlantic City. Mr. Offit pledged to line up cash for that, too.

Not long after, Edson Mitchell, a top bank executive, discovered that the signature of the credit officer who had approved the Trump Marina deal had been forged, Mr. Offit said. (Mr. Offit was never accused of forgery; the loan never went through.)

It was the first hiccup in the Trump relationship. It would not be the last.

The Mar-a-Lago Reward

Over the next few years, the commercial real estate group, with Mr. Kennedy now in a senior role, kept lending to Mr. Trump, including to buy the General Motors building in Manhattan. Occasionally, Justice Kennedy stopped by Deutsche Bank’s offices to say hello to the team, executives recalled.

At an annual pro-am golf tournament the bank hosted outside Boston in the early 2000s, Mr. Trump sat down for a recorded interview with the bank’s public relations staff, who asked about his experience with Deutsche Bank.

“It’s great,” Mr. Trump exclaimed, according to a person who witnessed the interview. “They’re really fast!”

In 2003, a Deutsche Bank team led by Richard Byrne — a former casino-industry analyst who had known Mr. Trump since the 1980s — was hired to sell bonds on behalf of Trump Hotels & Casino Resorts. Bank officials escorted Mr. Trump to meet institutional investors in New York and Boston, according to an executive who attended.

The so-called roadshow seemed to go well. At every stop, Mr. Trump was greeted by large audiences of fund managers, executives and lower-level employees eager to see the famous mogul. The problem, as a Deutsche Bank executive would explain to Mr. Trump, was that few of them were willing to entrust money to him.

Mr. Trump requested an audience with the bank’s bond salesmen.

According to a Deutsche Bank executive who heard the remarks, Mr. Trump gave a pep talk. “Fellas, I know this isn’t the easiest thing you’ve had to sell,” the executive recalled Mr. Trump saying. “But if you get this done, you’ll all be my guests at Mar-a-Lago,” his private club in Palm Beach, Fla.

The sales team managed to sell hundreds of millions of dollars worth of bonds. Mr. Trump was pleased with the results when a Deutsche Bank executive called, according to a person who heard the conversation.

“Don’t forget what you promised our guys,” the executive reminded him.

Mr. Trump said he did not remember and that he doubted the salesmen actually expected to be taken to Mar-a-Lago.

“That’s all they’ve talked about the past week,” the executive replied.

Mr. Trump ultimately flew about 15 salesmen to Florida on his Boeing 727. They spent a weekend golfing with Mr. Trump, two participants said.

A year later, in 2004, Trump Hotels & Casino Resorts defaulted on the bonds. Deutsche Bank’s clients suffered steep losses. This arm of the investment-banking division stopped doing business with Mr. Trump.

An Act of God

Around that time, Mr. Trump returned to Deutsche Bank’s commercial real estate unit — which was housed in a separate part of the sprawling investment-banking division — for another loan. This one was to build a 92-story skyscraper in Chicago, the Trump International Hotel and Tower.

Josef Ackermann, the bank’s chief executive, had publicly promised soaring profits, and with many of the company’s businesses sputtering, the investment-banking group was under intense pressure to grow.

As Deutsche Bank considered making the loan, Mr. Trump wooed bankers with flights on his private plane, according to a person familiar with the pitch. In a Trump Tower meeting, he told Mr. Kennedy that his daughter Ivanka would be in charge of the Chicago project, a sign of the family’s commitment to its success.

But there were warning signs.

Mr. Trump told Deutsche Bank his net worth was about $3 billion, but when bank employees reviewed his finances, they concluded he was worth about $788 million, according to documents produced during a lawsuit Mr. Trump brought against the former New York Times journalist Timothy O’Brien. And a senior investment-banking executive said in an interview that he and others cautioned that Mr. Trump should be avoided because he had worked with people in the construction industry connected to organized crime.

Nonetheless, Deutsche Bank agreed in 2005 to lend Mr. Trump more than $500 million for the project. He personally guaranteed $40 million of it, meaning the bank could come after his personal assets if he defaulted.

By 2008, the riverside skyscraper, one of the tallest in America, was mostly built. But with the economy sagging, Mr. Trump struggled to sell hundreds of condominium units. The bulk of the loan was due that November.

Then the financial crisis hit, and Mr. Trump’s lawyers sensed an opportunity.

A provision in the loan let Mr. Trump partially off the hook in the event of a “force majeure,” essentially an act of God, like a natural disaster. The former Federal Reserve chairman Alan Greenspan had called the financial crisis a tsunami. And what was a tsunami if not a natural disaster?

“It’s brilliant!” Mr. Schlesinger recalled Mr. Trump responding.

Days before the loan was due, Mr. Trump sued Deutsche Bank, citing the force majeure language and seeking $3 billion in damages. Deutsche Bank countersued and demanded payment of the $40 million that Mr. Trump had personally guaranteed.

With the suits in court, senior investment-banking executives severed ties with Mr. Trump.

Luring a Banker

Not long after Mr. Trump got the Chicago loan — but before it went south — Deutsche Bank was expanding its private-banking division, which served the superrich. Executives said they set out to hire Ms. Vrablic, whom they viewed as the best private banker in New York.

Traditionally, private bankers discreetly manage customers’ wealth and act as high-end concierges. Ms. Vrablic, who started her career as a bank teller and then worked at Citigroup and Bank of America, did that and more. She also arranged large real estate and commercial loans for her best clients.

To lure her, Deutsche Bank guaranteed that she would earn at least $3 million a year, unusually rich terms for a private banker, and would bypass a layer of management to report directly to Thomas Bowers, the head of the American wealth-management division, according to people familiar with her contract.

“Rosemary is widely recognized as one of the top private bankers to the U.S. ultra high-net-worth community,” Mr. Bowers said in a September 2006 news release. Deutsche Bank took out an ad in The Times to celebrate the arrival of her and a few colleagues.

‘There Is No Objection’

In 2010, Deutsche Bank and Mr. Trump settled their litigation over the Chicago loan. Mr. Trump agreed to repay most of what he owed by 2012, Mr. Schlesinger said.

One of Ms. Vrablic’s clients was Jared Kushner, who married Ivanka Trump in 2009. Mr. Kushner regarded Ms. Vrablic as the best banker he had ever worked with, according to a person familiar with his thinking.

Shortly after the Chicago lawsuit was settled, Mr. Kushner was told that Mr. Trump was looking for a loan and introduced him to Ms. Vrablic, according to people familiar with the relationship.

Mr. Trump flew Ms. Vrablic to Miami to show her a property he wanted to buy: the Doral Golf Resort and Spa. He needed more than $100 million for the 72-hole property.

Deutsche Bank dispatched a team to Trump Tower to inspect Mr. Trump’s personal and corporate financial records. The bankers determined he was overvaluing some of his real estate assets by as much as 70 percent, according to two former executives.

By then, though, Mr. Trump had become a reality-TV star, and he was swimming in cash from “The Apprentice.” Deutsche Bank officials also were impressed that Mr. Trump did not have much debt, according to people who reviewed his finances. Aside from his history of defaults, he was an attractive borrower.

Mr. Trump also expressed interest in another loan from the private-banking division: $48 million for the same Chicago property that had provoked the two-year court fight.

Mr. Trump told the bank he would use that loan to repay what he still owed the investment-banking division, the two former executives said. Even by Wall Street standards, borrowing money from one part of a bank to pay off a loan from another was an extraordinary act of financial chutzpah.

Ms. Vrablic and Mr. Bowers tentatively agreed to both loans.

Because these would be the private bank’s first transactions with Mr. Trump, they needed approval up the chain of command.

Investment-banking executives, including Anshu Jain, who would soon become Deutsche Bank’s co-chief executive, pushed back. Lending to Mr. Trump again would be foolish, they argued, and signal to clients that they could default and even sue the bank.

Executives in the private bank countered that the proposed loans had Mr. Trump’s personal guarantee and therefore were low risk. And the Chicago loan, they noted, would lead to the repayment of tens of millions of dollars that Mr. Trump still owed the investment-banking division.

A top executive with responsibility for the private bank discussed the loans with Mr. Ackermann, the chief executive, who supported them, according to two officials. A powerful committee in Frankfurt, which evaluated loans based on risks to the bank’s reputation, signed off.

“There is no objection from the bank to proceed with this client,” wrote Stuart Clarke, the chief operating officer for the Americas, in a Dec. 5, 2011, email, according to a recipient.

Deutsche Bank wired the money to Mr. Trump. The loans carried relatively low interest rates, executives said, but the business promised to be profitable: As part of the deal, Mr. Trump would hold millions of dollars in a personal account, generating fees for the bank.

“I have no recollection of having been asked to approve that private-banking loan,” Mr. Ackermann said in an interview. He added: “I would have approved it, if it came to me, if it was commercially sound.”

The C.E.O. Meets Trump

Ms. Vrablic’s relationship with the Trumps deepened.

Deutsche Bank lent money to Donald Trump Jr. for a South Carolina manufacturing venture that would soon go bankrupt. It provided a $15 million credit line to Mr. Kushner and his mother, according to financial documents reviewed by The Times. The bank previously had an informal ban on business with the Kushners because Jared’s father, Charles, was a felon.

In 2012, Jared Kushner recommended that the editor of The Mortgage Observer, one of the publications he owned, write a profile of Ms. Vrablic. The editor, Carl Gaines, knew Mr. Kushner was her client and objected, according to a person familiar with the exchange.

“Just go meet with her,” Mr. Kushner said. “You’ll figure something out.”

A gauzy profile of Ms. Vrablic was published in February 2013.

Shortly afterward, the private bank produced a promotional video featuring some of its marquee clients. The video was played at a retreat for Deutsche Bank’s senior leadership in Barcelona. In it, Ivanka Trump extolled the private bank’s work with her family and thanked their relationship manager, according to two people who saw the video.

In early 2014, Mr. Trump and his personal lawyer, Michael Cohen, approached Ms. Vrablic about more potential loans.

The owner of the Buffalo Bills had died, and the N.F.L. franchise was up for sale. Mr. Trump was interested, and he needed to show the league he had the financial wherewithal to pull off a transaction that could top $1 billion.

Mr. Trump asked Ms. Vrablic if the bank would be willing to make a loan and handed over bare-bones financial statements that estimated his net worth at $8.7 billion.

Mr. Cohen testified to Congress last month that the documents exaggerated Mr. Trump’s wealth. Deutsche Bank executives had reached a similar conclusion. They nonetheless agreed to vouch for Mr. Trump’s bid, according to an executive involved.

Mr. Trump’s bid did not win, but another lending opportunity soon arose.

A federal agency had selected Mr. Trump to transform the Old Post Office Building in Washington into a luxury hotel. But his financial partner — the private equity firm Colony Capital, run by Thomas J. Barrack Jr. — pulled out. Mr. Trump needed nearly $200 million.

Ms. Vrablic was willing to help.

In a memo outlining the rationale for the Old Post Office loan, Ms. Vrablic said Mr. Trump was expected to add large sums to his brokerage account if he received the loan, according to an executive who read the document.

This time, there was less internal opposition. One reason: Mr. Jain — by then the bank’s co-chief executive — had a solid relationship with Ms. Vrablic. Mr. Jain accompanied her to meetings with high-profile clients, and he praised her work to colleagues, multiple executives said.

On a foggy Wednesday in February 2013, Ms. Vrablic and Mr. Jain went to Trump Tower to meet with Mr. Trump, according to two executives with knowledge of the meeting. Ms. Vrablic’s rapport with the client was immediately clear: Mr. Trump’s assistant greeted her as an old friend, and she seemed relaxed with Mr. Trump and his daughter, one executive said.

They discussed Mr. Trump’s finances over lunch, and Mr. Jain said he was surprised by his low level of debt, the executives said. After lunch, Ms. Vrablic told her colleagues that Mr. Jain had sounded upbeat about Mr. Trump’s finances.

A $170 million loan to pay for the overhaul of the Old Post Office went through in 2015, and Mr. Trump added more money to his brokerage account. (In May 2016, he reported up to $46 million of stocks and bonds in the account.)

On Aug. 6, 2015, Mr. Trump participated in the first Republican presidential debate. He clashed with the Fox News moderator, Megyn Kelly. He flew back to New York early the next morning. That evening, he called in to a CNN talk show and said of Ms. Kelly that there was “blood coming out of her wherever.”

Shock and Anger

On the campaign trail, rivals assailed Mr. Trump’s financial history. In response, he pointed to Deutsche Bank-funded successes like the Old Post Office project, now a gleaming hotel a few blocks from the White House.

In early 2016, Mr. Trump asked Ms. Vrablic for one final loan, for his golf course in Turnberry, Scotland.

Ms. Vrablic said yes, but a fight soon erupted.

Jacques Brand, who was in charge of Deutsche Bank’s American businesses, angrily objected, partly because of Mr. Trump’s divisive rhetoric.

Ms. Vrablic appealed the decision. Senior executives in Frankfurt, including Christian Sewing, who would become chief executive in 2018, were shocked that the private bank would consider lending Mr. Trump money during the campaign, bank officials said.

The bank’s reputational risk committee killed the transaction in March 2016.

That same month, as The Times was preparing an article about Mr. Trump’s excommunication from Wall Street, he cited his warm relationship with Deutsche Bank.

“They are totally happy with me,” he said to The Times. “Why don’t you call the head of Deutsche Bank? Her name is Rosemary Vrablic. She is the boss.”

Demagogy and Defaults

After Mr. Trump won the election, Deutsche Bank’s board of directors rushed to understand how the bank had become the biggest lender to the president-elect.

A report prepared by the board’s integrity committee concluded that executives in the private-banking division were so determined to win business from big-name clients that they had ignored Mr. Trump’s reputation for demagogy and defaults, according to a person who read the report.

The review also found that Deutsche Bank had produced a number of “exposure reports” that flagged the growing business with Mr. Trump, but that they had not been adequately reviewed by senior executives.

On Deutsche Bank’s trading floor, managers began warning employees not to use the word “Trump” in communications with people outside the bank. Salesmen who violated the edict were scolded by compliance officers who said the bank feared stoking public interest in its ties to the new president.

One reason: If Mr. Trump were to default on his loans, Deutsche Bank would have to choose between seizing his assets or cutting him a lucrative break — a situation the bank would rather resolve in private.

Two years after Mr. Trump was sworn in, Democrats took control of the House of Representatives. The chamber’s financial services and intelligence committees opened investigations into Deutsche Bank’s relationship with Mr. Trump. Those inquiries, as well as the New York attorney general’s investigation, come at a perilous time for Deutsche Bank, which is negotiating to merge with another large German lender.

Ms. Vrablic, who is intensely private and rarely discusses her personal life with colleagues, declined to comment. People familiar with her thinking said she expected to be called to testify publicly on Capitol Hill.