President Biden’s looming decision about who should be the next Federal Reserve chairman is prompting reviews of the current chief’s record on bank regulation and how strict the postcrisis rulebook should be for Wall Street.

During Chairman Jerome Powell’s nearly four years as head of the Fed, the central bank has revamped big-bank stress tests, tailored its rules for U.S. lenders based on their size and simplified key postcrisis regulations such as the Volcker rule prohibition on proprietary trading.

Mr. Powell says that collectively the moves have clarified or better calibrated the central bank’s rules to reflect the risks posed to the financial system by the firms subject to them. In last year’s pandemic-driven, real-world stress test of the banking system, U.S. lenders emerged in solid financial shape, with stronger capital than before, he has said.

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“Strong capital requirements are essential for banks, particularly for the largest banks,” Mr. Powell told reporters Wednesday, at the conclusion of a two-day Fed policy meeting.

Some progressive Democrats say Mr. Powell’s Fed hasn’t been tough enough on large financial firms, arguing that the central bank’s tweaks to rules adopted after the 2008-09 financial crisis significantly softened the impact of the 2010 Dodd-Frank law, designed to ward off another financial crisis. Were it not for trillions in fiscal relief from Congress and an array of Fed backstops to credit markets, the banks might have gotten into trouble during the pandemic last year, the argument goes.

“The overall picture is that the financial regulatory safeguards have been materially eroded over the past four years,” said Gregg Gelzinis of the Center for American Progress, a center-left think tank that isn’t taking a position on whether Mr. Powell should be offered a second term as chairman.

At a July hearing, Sen. Elizabeth Warren (D., Mass.) slammed Mr. Powell’s record on easing regulation. Sen. Sherrod Brown (D., Ohio), chairman of the Senate Banking Committee, which holds confirmation hearings on Fed nominees, expressed similar criticism.

Mr. Powell’s four-year term as chairman expires in February, and both Ms. Warren and Mr. Brown have declined to say whether they would support him if he were offered a second term.

Mr. Powell, a Republican, is viewed by some inside and many outside the administration as the front-runner for the job, but isn’t a lock. Fed governor Lael Brainard, a Democrat, has emerged as the most likely candidate to succeed him should Mr. Biden decide he would prefer his own pick rather than the leader chosen by then-President Donald Trump.

Ms. Brainard, an important ally of Mr. Powell on monetary policy, has regularly dissented against his decisions to ease bank regulations enacted after the 2008-09 financial crisis, which is unusual at an institution that historically supervises banks through consensus. Ms. Brainard has said generally that the Fed’s moves to soften regulations have gone too far, weakening core safeguards against vulnerabilities that led to the financial crisis.

Lael Brainard, a Federal Reserve governor, has regularly dissented against decisions to ease bank regulations put in place after the 2008-09 financial crisis.

Lael Brainard, a Federal Reserve governor, has regularly dissented against decisions to ease bank regulations put in place after the 2008-09 financial crisis.

Photo: Andrew Harrer/Bloomberg News

A White House official said recently, “The president will engage with his senior economic team in a careful and thoughtful process to appoint a Federal Reserve chair in a timely manner.” The White House declined to comment on specific names that may be under consideration.

White House officials haven’t said publicly how much Mr. Powell’s regulatory record will weigh in deciding whether to offer him a second term as chairman.

After the 2008-09 financial crisis, the credibility of financial firms took a big hit among regulators and lawmakers of both parties, leading to stricter rules for the industry and less influence over Washington’s decisions about how the firms ran their businesses. Mr. Powell, who was first nominated in 2011 by then-President Barack Obama to the Fed’s Board of Governors, supported those restrictions before being tapped by Mr. Trump to become chairman in 2018.

Federal Reserve Chairman Jerome Powell in June described the outlook for inflation in the U.S. economy and said there are signs that prices that have moved up quickly should cease rising and retreat. Credit: Al Drago/Associated Press The Wall Street Journal Interactive Edition

Though Mr. Trump promised to dismantle the Dodd-Frank financial overhaul, he ultimately settled on a less-ambitious objective: legislatively raising the asset threshold at which regional lenders automatically faced stricter rules. Major planks of Dodd-Frank remained unchanged, such as emergency government powers and curbs on derivatives, cementing those provisions for years to come.

While banks won some regulatory victories from the Fed and other regulators during the Trump administration, such as eased margin requirements for certain swap transactions, they struck out on priorities such as a reduction in capital requirements at the biggest firms. Industry-backed efforts to ease a capital surcharge on the largest U.S. banks went nowhere during Mr. Powell’s tenure. Similarly, the Fed declined to support an overhaul of lower-income lending standards favored by another Trump-appointed regulator but opposed by Democrats and consumer advocates.

Under then-Chairwoman Janet Yellen, the Fed put limits on Wells Fargo & Co.’s size as punishment for its 2016 fake-account scandal. Mr. Powell left the cap in place, except for a small period last year, when the Fed temporarily lifted the restrictions so the bank could make loans through two federal small-business lending programs during the coronavirus crisis.

Mr. Powell also moved ahead with a proposal to develop a faster payments system for banks to exchange money, something opposed by big banks that have built a separate network.

And while the bank’s annual stress tests have become less stressful in some ways—through the elimination of what had historically been pass-fail grades, for instance—some changes to the tests were endorsed by former Fed governor Daniel Tarullo, a Democrat who served as the central bank’s pointman on regulation from 2009 until 2017.

“The narrative that Chairman Powell oversaw a sweeping rollback of the regulatory regime is hyperbole at best,” said Isaac Boltansky, director of policy research at Compass Point Research & Trading. “There’s progressive unhappiness with the regulatory softening we saw during the Trump administration, but the actions in question were modest, didn’t have a material impact on financial stability, and shouldn’t be the sole determinant for Powell’s prospects.”

Ms. Warren chastised Mr. Powell at the hearing for what she described as repeated steps to ease the Volcker rule trading restrictions and allow banks to submit full versions of orderly wind-down plans known as “living wills” less frequently.

She said Mr. Powell had overseen a long list of regulatory rollbacks. “Reducing capital requirements, easing liquidity requirements, shrinking margin requirements, scaling back on supervision,” she said. “Can you name a change that strengthened the rules and made the actual rules tougher?”

Mr. Powell denied that the Fed weakened capital requirements for the largest banks. “I actively resisted any move in that direction,” he said. A so-called stress capital buffer—a new framework to incorporate banks’ annual “stress test” results into their capital requirements for the ensuing year—had the effect of raising the largest firms’ capital requirements, according to the Fed.

Write to Andrew Ackerman at andrew.ackerman@wsj.com