“Across a range of responsibilities, we simply expect much more of boards of directors than ever before,” Mr. Powell said in a speech in August. “There is no reason to expect that to change.”

This account is based on interviews with six people involved in or briefed on the negotiations, representing both the bank and the Fed, who were not authorized to speak publicly about regulatory matters.

Wells originally got into trouble in 2016 for charging millions of customers for bank accounts they did not want and for auto insurance they did not need. The bank was repeatedly penalized and fined by regulators.

Executives had convinced themselves last year that they were out of the woods, according to the people familiar with their thinking, who were not authorized to speak publicly about interactions with regulators. But that illusion was shattered in September, when Ms. Yellen said the bank remained under investigation.

In early January, Wells officials heard from the Fed that the central bank planned to impose stiff new penalties. Executives were furious that the proposed sanctions seemed more draconian than those imposed on banks that nearly cratered the global economy a decade earlier, according to people familiar with the thinking of top bank executives.

Then, on conference calls and face-to-face sessions in Washington, the negotiations began.

One crucial participant was Wells’s general counsel, C. Allen Parker. His advantage was that he was new to Wells, not part of what one bank adviser called the “ancien régime.” He joined last spring after more than two decades at Cravath, Swaine & Moore, the white shoe law firm where he had been the presiding partner.