The issue could come to a head within weeks. When the House reconvenes on April 25, the first order of business will be a spending bill to replace the current stopgap law, which expires three days later. Democrats are determined to put money for the health insurance subsidies into that bill, and some Republicans on the House and Senate Appropriations Committees are open to the idea. But ultimately, the decision will be made by Republican leaders in the two chambers.

If the spending is allowed to continue, the Congressional Budget Office estimates that the federal government will pay $135 billion in cost-sharing subsidies to insurers from 2018 to 2027.

The cloud of uncertainty swirling around the subsidies stems from a court ruling in a lawsuit that House Republicans filed against the Obama administration in 2014. Judge Rosemary M. Collyer of the Federal District Court in Washington ruled last year that spending on the subsidies “violates the Constitution” because Congress never appropriated money for them. She ordered a halt to the payments, but suspended her order to allow the government to appeal.

The Trump administration has not made clear whether it will press the appeal filed by the Obama administration. In a letter to Mr. Trump this week, the U.S. Chamber of Commerce joined the American Medical Association, the American Hospital Association and insurers in seeking “quick action” to guarantee continuation of the subsidies. Without the subsidies, they said, more people will be uninsured and unable to pay medical bills.

Democrats say they will not negotiate with Mr. Trump until he stops his drive to repeal the Affordable Care Act. “President Trump is threatening to hold hostage health care for millions of Americans, many of whom voted for him, to achieve a political goal of repeal that would take health care away from millions more,” said the Senate Democratic leader, Chuck Schumer of New York.

Having no immediate prospect of a deal with Democrats, the White House is still focusing on Republicans. It is seeking consensus on a repeal bill that can overcome rifts among House Republicans, whose disagreements sank an earlier version of the legislation on March 24.

Even though lawmakers are out of town for a two-week spring break, Mr. Trump and Vice President Mike Pence are continuing efforts to revive the bill in talks with Representative Mark Meadows, the North Carolina Republican who is the chairman of the conservative House Freedom Caucus.