“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.