Senator Elizabeth Warren says Wells Fargo has run out of time to fix the many internal problems that have harmed its customers.

In a letter to the Federal Reserve chair, Jerome H. Powell, on Monday, Ms. Warren asked the Fed to force the financial giant to break off its core banking activities, like offering checking and savings accounts and loans, from its other financial services.

Divorcing Wall Street-centric work — which can include managing investment funds and providing financial market sales and trading services — from the bank would ensure that Wells Fargo’s everyday customers did not continue to suffer, Ms. Warren wrote. The Fed could accomplish this, she explained, by revoking Wells Fargo’s financial holding company license — essentially making it impossible for the company to operate any nonbanking businesses.

“Continuing to allow this giant bank with a broken culture to conduct business in its current form poses substantial risks to consumers and the financial system,” she wrote.

It’s the first time that Ms. Warren, a Massachusetts Democrat, has made such a request to a regulator. Mr. Powell is unlikely to follow the suggestion, at least in the near future: Wells Fargo would somehow have to quickly shed dozens of nonbank subsidiaries, which would certainly undermine the bank’s financial health.

But Ms. Warren’s request for such aggressive action hinted at the desire among progressives for a stronger regulatory hand from the Fed. And it comes as President Biden is considering whether he will reappoint Mr. Powell when his term ends early next year.

A Fed spokesman confirmed that the central bank had received the letter.

A Wells Fargo spokeswoman responded to a request for comment by emailing journalists a list of steps the bank had taken since 2019 to change its management structure. The list included replacing top managers, splitting businesses into new groups with separate oversight, creating a “sales practices oversight” protocol and speeding up restitution to customers it harmed.

Wells Fargo has spent years trying to right its standing with regulators and lawmakers after a cascade of disclosures of misconduct by the bank against its customers. It admitted to opening accounts in their names without their knowledge, forcing them to buy unnecessary insurance and charging them unwarranted mortgage fees.

The bank’s statement also pointed out that Wells Fargo had fulfilled two requirements for improvement that regulators had placed on it. One, from 2015, forced it to improve its anti-money-laundering oversight. The other, from 2016, required it to end harmful sales practices.

The idea of revoking the bank’s financial holding company license was first proposed two years ago by Jeremy Kress, an assistant professor of business law at the University of Michigan’s Ross School of Business. He argued in an academic paper that, since the Fed requires institutions seeking such a license to have a high regulatory rating — a 1 or 2 out of 5 on a scale that federal regulators use to assess whether a bank is well managed — any bank whose rating subsequently sinks lower should lose it.

In her letter, Ms. Warren pointed to a report from early 2018 revealing that Wells Fargo’s regulatory rating, normally kept secret, had fallen below the level at which the bank could be considered “well managed.” She added that it was “inconceivable” that Wells Fargo could have recently improved its rating given its continuing problems.

Last week, federal regulators announced another set of fines and restrictions on the bank, stemming from its inappropriate handling of some of its home loan customers’ portfolios. The Office of the Comptroller of the Currency found that Wells Fargo’s management of its mortgage accounts had been so sloppy that it might have improperly foreclosed on some borrowers’ homes. The regulator fined the bank $250 million, ordered it to halt some foreclosures in progress and gave it five months to get its management systems on track.

Ms. Warren’s suggestion would be a much more significant penalty and would involve significant challenges, said Jaret Seiberg, an analyst for Cowen Washington Research Group.

“The banking system has evolved considerably since Congress allowed investment banks and commercial banks to get together 25 years ago,” Mr. Seiberg said. “These operations are now interwoven, and it’s not as simple as saying, ‘You have to sell half the bank.’ Breaking up is hard to do.”

But immediately revoking the bank’s license is not Mr. Powell’s only option.

In an interview on Tuesday, Professor Kress said Mr. Powell could direct Fed officials to issue a new rule setting a limit — two or three years, for example — on the length of time a company could operate with a low management rating and maintain its license.

“It could set a precedent that this is a tool that the Fed can use to hold banks accountable for financial misconduct,” Mr. Kress said.

The Fed has already taken drastic measures to try to force Wells Fargo to improve. Since early 2018, the bank has been operating under an asset cap, which the Fed vowed to keep in place until the bank could prove it had overhauled its risk-management procedures and established better protections for its customers. But Ms. Warren said the bank was distracted from that goal, citing reports that Wells Fargo was trying to expand activities like putting together corporate mergers and other investment banking services.

The bank should be forced to give up those Wall Street pursuits “to ensure that its leaders focus all of their attention on fixing the bank’s numerous, chronic risk-management deficiencies,” she wrote.

Wells Fargo is the country’s fourth-largest bank, though its Wall Street presence — including investment banking and wealth management services — is much smaller than those of competitors like JPMorgan Chase and Bank of America. Its chief executive, Charles W. Scharf, has a Wall Street background and, since taking over two years ago, has tried to make Wells Fargo more profitable by steering it more toward Wall Street.

“I am concerned that Wells Fargo’s senior executives are focused on expanding risky investment banking activities instead of remediating consumer harms and improving lax internal controls,” Ms. Warren said, though her letter to Mr. Powell did not mention Mr. Scharf by name.

Ms. Warren sent a separate letter on Monday to the chairman of Wells Fargo’s board of directors. It asked for details about how the board is overseeing the bank’s cleanup efforts and why it is paying Mr. Scharf so well — he received more than $20 million in the 2020 fiscal year — even as the bank’s problems endure.