Lawmakers also said there was increasing talk of Mr. Biden invoking the 14th Amendment of the Constitution to raise the debt ceiling unilaterally, a move they acknowledged would draw a legal challenge — and which Ms. Yellen has questioned — but could still avert economic disaster.

With so much uncertainty, Senate Democrats were also weighing whether they would be able to take a weeklong recess scheduled to begin on Monday, before the Memorial Day weekend.

Some potential areas of compromise have emerged in recent days. Mr. McCarthy said on Monday that he wanted to negotiate some of the key provisions of the bill to raise the debt limit that House Republicans passed last month. Those include spending caps, permitting changes for domestic energy projects, work requirements for safety net programs like food stamps and clawing back unspent money allocated for pandemic relief programs.

“All of that I felt would be very positive,” Mr. McCarthy said.

In addition to the speaker, Senator Mitch McConnell, Republican of Kentucky and the minority leader; Senator Chuck Schumer, Democrat of New York and the majority leader; and Representative Hakeem Jeffries of New York, the Democratic leader, will join Mr. Biden at the White House.

The government reached the $31.4 trillion debt limit on Jan. 19, and the Treasury Department has been using accounting maneuvers to keep paying its bills. The president is scheduled to depart for Japan on Wednesday to attend the Group of 7 meeting, heightening the sense of urgency to make progress on the debt limit.