“I’m looking at the 14th Amendment, as to whether or not we have the authority,” he said during a news conference with reporters in Hiroshima, Japan, over the weekend. “I think we have the authority. The question is: Could it be done and invoked in time that it could not — would not be appealed and, as a consequence, pass the date in question and still default on the debt.”

Mr. McCarthy sounded more upbeat on Sunday after the call with Mr. Biden, in which the two pledged to meet. The president “walked through some of the things that he’s still looking at, he’s hearing from his members; I walked through things I’m looking at,” Mr. McCarthy said. “I felt that part was productive. But look — there’s no agreement. We’re still apart.”

Chief among the outstanding issues is how much to spend overall next fiscal year on discretionary programs and how long any spending caps should be in place. The latest White House proposal would hold both military and other spending — which includes education, scientific research and environmental protection — constant from the current fiscal year to the next. Republicans have insisted that overall discretionary spending drop as they still call for increased military spending.

The number of legislative days for Congress to vote to raise the debt ceiling before the projected June 1 deadline is rapidly dwindling. Treasury Secretary Janet L. Yellen said in an interview with NBC News over the weekend that the odds of the government getting to mid-June before defaulting were “quite low.”

Once negotiators agree to a deal, it will take time to translate it into legislative text. Mr. McCarthy has promised that he will give lawmakers 72 hours to review the bill.