President Biden and Speaker Kevin McCarthy expressed optimism on Monday that they could break the partisan stalemate that has prevented action to avert a default on the nation’s debt, but remained far apart on a deal to raise the debt limit as Democrats resisted Republicans’ demands for spending cuts in exchange.

The two met face to face at the White House for the second time in two weeks in a show of good will after a weekend of behind-the-scenes clashes among negotiators, punctuated by a move by Republicans on Friday to halt the talks and accusations by both sides that the other was being unreasonable.

With Mr. Biden back from a summit meeting in Japan, the tenor appeared to have changed considerably.

“We don’t have an agreement yet,” Mr. McCarthy told reporters at the White House after the meeting. “But I did feel like the discussion was productive,” he said, adding later that he believed the tone of the talks was “better than any other time we’ve had discussions.”

“I believe we can still get there,” Mr. McCarthy said. “I believe we can get it done.”

He said he expected to speak with Mr. Biden daily until a deal could be struck.

With a default looming as soon as June 1, both Mr. Biden and Mr. McCarthy began their latest meeting sounding upbeat about finding common ground in an effort to avoid economic catastrophe and left dispatching their top advisers to hammer out an agreement in the coming days.

“We still have some disagreements, but I think we may be able to get where we have to go,” Mr. Biden said as the two sat down in the Oval Office. “We both know we have a significant responsibility.”

Mr. Biden said in a brief statement after the meeting that the talks were “productive.”

“We reiterated once again that default is off the table and the only way to move forward is in good faith toward a bipartisan agreement,” he added, saying that he and his negotiating team would continue talking with Mr. McCarthy and his.

Still, the two sides remained at loggerheads. The White House has called Republicans’ demands for spending cuts extreme, while Mr. McCarthy and his aides have accused White House officials of being unreasonable.

The number of legislative days for Congress to vote to raise the debt ceiling before the projected deadline is rapidly dwindling. Treasury Secretary Janet L. Yellen on Monday reiterated her warning to Congress that the United States could exceed its authority to borrow to pay its bills as soon as June 1. She said in an interview with NBC’s “Meet the Press” over the weekend that the odds of the government being able to hold out until mid-June — when a substantial amount of quarterly tax revenue is expected to roll in, giving the Treasury more breathing room to cover its obligations — were “quite low.”

And Republicans hinted that no deal was likely to materialize until a default was truly imminent. When asked on Monday evening what it would take to break the deadlock, Mr. McCarthy replied simply: “June 1.”

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. Republicans want to allow military spending to increase while cutting other programs. But they have shown some flexibility around how long they would seek to cap spending overall, coming down from their initial demand of a decade to six years.

That is longer than Mr. Biden wants. White House officials have proposed holding both military and other spending — which includes education, scientific research and environmental protection — constant over the next two years.

“These are tough issues,” said Representative Patrick T. McHenry, Republican of North Carolina and a key ally of Mr. McCarthy who has been involved in the talks and attended the White House meeting. “A directive to cut spending year over year is the toughest thing to do in Washington, D.C. But that is the speaker’s directive to his negotiating team. It is our expectation to be able to get that.”

Hard-right members of Mr. McCarthy’s conference have continued to pressure the speaker not to accept anything less than the spending cuts that House Republicans passed in their debt limit bill last month, which would have amounted to a reduction of an average of 18 percent over a decade.

“Republicans must #HoldTheLine on the debt ceiling to bring spending back to reality and restore fiscal sanity in DC,” the House Freedom Caucus wrote on Twitter. “We spend $100+ billion more than federal tax revenues EVERY MONTH. Washington has a spending problem, not a revenue problem.”

Mr. McCarthy expressed confidence that he could keep his conference largely united around whatever deal he strikes with Mr. Biden, telling reporters at the Capitol before the meeting that he believed it would draw the support of both Democrats and Republicans.

“I firmly believe what we’re negotiating right now, a majority of Republicans will see that it is a right place to put us on a right path,” he said.

But he also hinted that members of his conference should prepare to accept a final product that falls short of what some lawmakers have demanded.

“I don’t want you to think at the end of the day, the bill that we come up with is going to solve all this problem,” he said. “But it’s going to be a step to finally acknowledge our problem and put one step in the right direction. And we’re going to come back the next day and get the next step.”

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, and said on Monday that he believed negotiators would need to agree to a compromise this week in order to pass legislation raising the debt ceiling before the projected June 1 deadline.

Lawmakers in the House were still left uncertain about when they would need to be present to cast a vote to avert a default. The House, as of Monday evening, was set to depart Washington beginning on Thursday afternoon ahead of the Memorial Day weekend.

The two sides have found some agreement in talks in the past week, including on clawing back some unspent funds from previously approved Covid-19 relief legislation.

Senior administration officials said Project NextGen, the Biden administration’s $5 billion Covid vaccine development program, could be among the casualties of those cuts. The program, modeled in part on the Trump administration’s Operation Warp Speed, is an effort to find different forms of vaccines that scientists believe will offer more durable protection against the coronavirus.

But many other issues have yet to be resolved, including tightening work requirements for able-bodied adults without dependents for certain safety social net programs. The bill passed by House Republicans contained stricter requirements for recipients of Temporary Assistance for Needy Families and food stamps, and is a key demand of conservatives in the House.

Mr. McCarthy said on Monday that he would continue to push for their inclusion in whatever deal he strikes with Mr. Biden, and White House negotiators have shown openness to finding some compromise on the issue.

Carl Hulse contributed reporting.