Under pressure from House Democrats and Republicans on Thursday to take a position, Ms. Yellen wouldn’t say whether the Biden administration favors repealing the $10,000 cap on the deduction. Eliminating that limit would cut taxes for high-income households and disproportionately benefit residents of high-tax Democratic-leaning states such as New York, New Jersey and California.

Mr. Biden’s budget is silent on the deduction cap, which was created in the 2017 tax law and is scheduled to lapse after 2025, along with many of that law’s other provisions.

“He recognizes that it’s a tremendous concern in a number of states and wants to work with Congress to see if there’s a way to mitigate the harms that it’s caused,” Ms. Yellen said of the president during a Ways and Means Committee hearing. “But he has not made a proposal in the budget to do so.”

Rep. Kevin Brady

(R., Texas), the committee’s top Republican, pressed Ms. Yellen on whether Mr. Biden would support the cap’s elimination if lawmakers included it in a compromise infrastructure package. “I’m not going to negotiate here on behalf of the president,” she said.

The deduction’s fate may be crucial to getting the Biden agenda through the House. Some lawmakers, including Reps.

Tom Suozzi

(D., N.Y.) and Bill Pascrell (D., N.J.) are pushing for the cap’s repeal as part of any tax legislation that moves. Democrats have a slim majority in the House, and just a few defections could imperil a tax bill.

But some progressives oppose repeal because so much of the benefits go to the top sliver of households. About half the benefits of repeal would go to households making more than $1 million, according to the Tax Policy Center, a project of the Urban Institute and Brookings Institution.

Some lawmakers have suggested raising the cap instead of repealing it. That move would help many upper-middle-class households in high-cost areas. But it wouldn’t address another concern of state officials—the idea that the lack of a deduction would make it harder for them to raise taxes on very high-income households. With the cap, state income taxes come on top of federal income taxes, creating a larger incentive for people to save taxes by moving to Florida, Texas or other places without income taxes.

“We may not get the full relief. I know that,” Mr. Pascrell said to Ms. Yellen during the hearing. “But the administration’s silence on this makes my constituents worried.”

Republicans created the cap in 2017 to help pay for tax cuts for individuals, including lower rates, larger child tax credits for individuals and the narrowing of the alternative minimum tax that hit higher-income households in those same states. Most households—even in high-tax states—pay less in taxes than they did before the 2017 law.

Write to Richard Rubin at richard.rubin@wsj.com and Kate Davidson at kate.davidson@wsj.com