“The coming weeks are critical for assessing the strength of government cash flows,” said Shai Akabas, the director of economic policy at the Bipartisan Policy Center. “If a solution is not reached before June, policymakers may be playing daily Russian roulette with the full faith and credit of the United States, risking financial disaster for their constituents and the country.”

A default could come sooner than expected because tax revenues have been trickling into the government’s coffers this spring. The sluggish pace is due in part to a decision by the Internal Revenue Service to give taxpayers in states that were affected by severe weather more time to file their 2022 taxes.

The brinkmanship has renewed questions about how the federal government might try to prioritize certain payments if it does run out of cash, whether Mr. Biden could ignore the debt limit entirely and order the Treasury Department to continue borrowing, and if far-fetched ideas such as minting a $1 trillion coin could in fact be viable.

Treasury Secretary Janet L. Yellen said on Monday that if the debt limit was not raised, then Mr. Biden would have to decide how to proceed.

“I would say that if Congress doesn’t raise the debt ceiling, the president will have to make some decisions about what to do with the resources that we do have,” Ms. Yellen said on CNBC. “And there are a variety of different options, but there are no good options.”