The longtime chief executive of Wells Fargo agreed on Tuesday to forfeit $41 million in performance pay three weeks after the bank acknowledged that for at least five years, thousands of low-level employees set up sham accounts to meet sales quotas.

The San Francisco-based bank has repeatedly apologized for the scheme and said it had fired 5,300 employees for misconduct and put in place more stringent internal controls. But that has not been enough for lawmakers, who have been pushing for the company’s top leaders to give back the millions of dollars in bonuses they earned while the irregularities were occurring.

The independent directors of the Wells Fargo board announced Tuesday that they were launching an investigation into the bank’s retail business.

“We are deeply concerned by these matters, and we are committed to ensuring that all aspects of the Company’s business are conducted with integrity, transparency, and oversight,” Stephen Sanger, the lead independent director, said in a statement. Sanger, who has been a member of Wells Fargo’s board since 2003, is the former chairman of General Mills, the packaged-food company.

John Stumpf, the chief executive, will forfeit unvested stock awards worth about $41 million, will not receive a salary while the investigation is underway and will not be eligible for a 2016 bonus, the committee said in a statement. Carrie Tolstedt, the former head of Wells Fargo’s community banking unit, where the misconduct took place, will give up $19 million in unvested stock awards and not be eligible for a 2016 bonus. Tolstedt had announced her retirement in July but had initially planned to stay at the bank until the end of the year.

In a tense exchange, Senator Elizabeth Warren badgered Wells Fargo CEO John Stumpf on why he had not offered to give up any of his compensation or to resign in the wake of the fake accounts controversy. (Reuters)

Stumpf and Tolstedt still have millions of dollars in stock and other compensation from Wells Fargo. Stumpf, for example, was awarded $161 million in bonuses and performance awards between 2011 and 2015, the period in which the misconduct took place, according to Equilar, a research firm. Tolstedt was awarded more than $60 million in bonuses in the same period.

But this action by the bank’s board is, by far, the most aggressive and public effort by a bank since the 2008 financial crisis to show that top executives will be held responsible for their companies’ misdeeds.

In early September, Wells Fargo was fined $185 million by regulators after it discovered that thousands of employees were setting up unauthorized accounts, including credit cards and checking accounts, that customers had not requested. In some cases, the customers were charged various fees for accounts they did not know existed.

The case has sparked a national outcry, with lawmakers pummeling Stumpf before the Senate Banking Committee last week, with one even calling on him to resign. He is to be the only witness before the House Financial Services Committee when he attends a hearing this week, and the move to take back financial awards could help head off some of the inevitable anger of lawmakers.

The debate comes as regulators try to finalize rules called for by the 2010 financial-reform bill, known as Dodd Frank, to overhaul the way Wall Street executives are paid and to address years of complaints that excessive bonuses helped lead to the 2008 financial crisis.

The long-awaited rules were aimed at stopping executives from making risky financial bets to boost their pay and then collecting large bonuses before the consequences of their actions are clear. But the banking industry is mounting a counterattack against the rules, which critics say do not allow enough flexibility. Some employees would be subject to claw-backs for up to 11 years after being awarded bonuses, much longer than the current industry standard of three to five years, industry officials have said.

In July, Wells Fargo joined the debate, saying it was concerned that the rules may have unintended consequences. “We have worked, and are continuing to work, diligently . . . to improve and evolve our incentive-based compensation practices,” Hope Hardison, Wells Fargo’s director of human resources, said in a letter to regulators.

But the rules would “create an un-level playing field, driving high-quality talent” away from big banks subject to the rules, she wrote.

The controversy surrounding Wells Fargo is likely to make it more difficult for banks to mount efforts to weaken the rules, industry officials say.

Stumpf is expected to face tough questioning when he appears before the House Financial Services Committee on Thursday. In addition to questions about executive compensation, he is likely to face questions about a recently launched Labor Department inquiry into the bank.

The Labor Department says it is conducting a “top-to-bottom review” of Wells Fargo after lawmakers complained that the bank’s workers were forced into a hypercompetitive environment that led thousands of them to set up the sham accounts to meet aggressive sales goals.

Some former employees allege that they were unfairly fired when they did not cut corners to meet Wells Fargo’s aggressive sales targets.

That could violate federal rules, according to lawmakers and some labor experts. Last week, former Wells Fargo employees who say they were fired for following the law filed a $2.6 billion lawsuit against the bank. They are seeking class-action status.

The Labor Department said it has set up a working group to expedite a review of the issue.

“Given the serious nature of the allegations . . . I have directed enforcement agencies within the Department to conduct a top-to-bottom review of cases, complaints, or violations concerning Wells Fargo over the last several years,” Labor Secretary Thomas Perez said in a letter released Tuesday by Sen. Elizabeth Warren (D-Mass.), who had asked the agency to look into the matter.

“We take the concerns raised . . . very seriously,” Perez wrote.

A Wells Fargo spokeswoman declined to comment on the Labor Department review.

It is just the latest sign that San Francisco-based bank will be grappling with the fallout from this case for some time. Federal prosecutors are considering civil or criminal charges against the bank.