President Biden, Speaker McCarthy, and top House Republicans have met for their third round of debt ceiling negotiations as fears of a default abound. Little progress was made on concrete measures to curb the debt and future deficits, but Messrs. Biden and McCarthy struck a cordial tone that could bear fruit in the coming days. 

“I just concluded a productive meeting with Speaker McCarthy about the need to prevent default and avoid a catastrophe for our economy,” Mr. Biden said in a statement issued Monday evening. “While there are areas of disagreement, the Speaker and I … will continue to discuss a path forward.”

“I thought the meeting was productive,” the speaker told reporters on Capitol Hill after returning from the White House. “We still have differences. We left the meeting with directing … the staff to get back together, work through the night, knowing where our differences lie and see if there are other ideas” about how to move forward. 

Messrs. Biden and McCarthy previously met on May 16 and May 9 for debt ceiling negotiations after top aides worked through outlines for a potential compromise. 

On May 16, the two men sat down at the White House along with the minority leaders of the House and Senate, Congressman Hakeem Jeffries and Senator McConnell. After their meeting, Mr. McCarthy told reporters outside of the White House that the only agreement he reached with the administration is that each group designate a “point person” to lead negotiations for the coming weeks. 

Mr. McCarthy confirmed that he and the White House are “very far apart” even after Tuesday’s negotiating session. When asked if he is optimistic about the prospects for a deal, the speaker said he “would be” had the negotiations begun back in February, when Mr. Biden refused to sit down. 

Before the latest meeting, Mr. McCarthy had demanded that the White House acquiesce to work requirements for welfare — something Mr. Biden supported as a senator during President Clinton’s welfare reform efforts in the 1990s. 

In a sign of how worried the White House is about a potential default, Mr. Biden abandoned part of a foreign trip in order to be in Washington for negotiations ahead of the June 1 possible default date. The president returned from a G7 meeting in Japan on Sunday after canceling a scheduled visit to Papua New Guinea and Australia.

One of the greatest challenges for Mr. McCarthy is not the president’s intransigence, but rather the politics of managing the most conservative members of his caucus. As he fought to obtain the speaker’s gavel in January, Mr. McCarthy handed the ultraconservative Freedom Caucus a major victory: Any one member is now able to go to the House floor and call for a motion to vacate the speaker’s chair, meaning a single Republican who is dissatisfied with the speaker’s debt ceiling compromises could conceivably move to throw him out of the job.   

One Freedom Caucus member, Congressman Eli Crane, told CNN that he has had conversations with some of his caucus colleagues about calling for a vote of no confidence in the speaker, which could lead to another seemingly endless voting process to either retain Mr. McCarthy or choose a new leader.     

“It does come up from time to time,” Mr. Crane said of the possibility of removing Mr. McCarthy. “We look at all of the alternatives and contingency plans that could play out over the next two years.”

There is some hope for Mr. McCarthy, though, should he choose to make a deal that is palatable to Democrats. On Tuesday, Politico reported that a number of centrist House Democrats are open to protecting Mr. McCarthy from his right flank and keeping his job should he find a compromise. 

“We’ll protect him if he does the right thing,” one Democrat said of the speaker, signaling they would vote for him for speaker should a conservative lawmaker issue a motion to vacate the chair.