President Biden and congressional leaders will resume face-to-face talks on Tuesday to avert a government default, with the White House expressing cautious optimism as the contours of a possible deal began to come into focus.

With time running out to strike a deal to raise the debt limit, broad areas of negotiation have emerged, including fixed caps on federal spending, reclaiming unspent funds designated for the Covid-19 emergency, stiffer work requirements for federal benefits and expedited permitting rules for energy projects.

“I remain optimistic because I’m a congenital optimist,” Mr. Biden told reporters on Sunday in Rehoboth Beach, Del. He added, “I really think there’s a desire on their part, as well as ours, to reach an agreement, and I think we’ll be able to do it.”

Still, on Monday, Speaker Kevin McCarthy reiterated that he believed little progress had been made, telling reporters that the two sides remained “far apart” even with a potential default looming. “We have no agreements on anything. That’s why I’m so concerned,” he added.

Treasury Secretary Janet L. Yellen reiterated on Monday that the United States might be unable to pay its bills by June 1 if it does not raise or suspend the debt limit, which caps how much money the country can borrow.

That $31.4 trillion limit was hit on Jan. 19, and the Treasury Department has been using accounting maneuvers to keep paying the government’s bills. In a letter to lawmakers on Monday, Ms. Yellen cautioned that the actual date “might be a number of days or weeks later than these estimates” but she urged Congress to move quickly to prevent a default.

The Treasury Department has been using accounting maneuvers known as extraordinary measures to keep paying the country’s bills without breaching the debt ceiling.

Republicans have said they want to cut federal spending before lifting the ceiling, but Mr. Biden has maintained that negotiating over cuts must not be a condition for raising the limit and avoiding what might be a catastrophic default.

Economists on Wall Street and in the White House say a prolonged default might obliterate jobs and lead the country into a recession.

Mr. Biden, who is set to depart on Wednesday for Japan to attend the Group of 7 meeting, confirmed on Monday that he would meet with Mr. McCarthy on Tuesday. The meeting will be at 3 p.m., according to the White House.

Senator Chuck Schumer of New York, the majority leader, was more optimistic than Mr. McCarthy on Monday, saying that the “parallel discussions” on federal spending and the debt ceiling were continuing in “a very serious way.”

“We welcome a bipartisan debate regarding our nation’s fiscal future,” Mr. Schumer said. “But we’ve made it plain to our Republican colleagues that default is not an option. Its consequences are too damaging, too severe. It must be taken off the table.”

The two sides had their first face-to-face meeting at the White House last Tuesday, but it ended without a deal. They had been set to meet once more on Thursday, but that session was postponed to allow staff members more time to speak in detail.

People familiar with the negotiations cast the decision to postpone that meeting as a positive development, one that would give staff members more time to make progress.

“The conversations are constructive between all of the parties,” said Wally Adeyemo, the deputy Treasury secretary.

“The United States has never defaulted on its debt, and we can’t,” Mr. Adeyemo said. “Because defaulting on our debt isn’t just regarding financial markets. It’s regarding paying our Social Security recipients. It’s regarding paying our troops. It’s regarding paying the men and women who are working the border today.”

Biden administration officials have said they will not accept any deal that rolls back the president’s signature legislative achievements, particularly on climate change. They want Republicans to drop certain provisions in the debt limit bill that passed the House last month.

That measure is dead on arrival in the Democratic-led Senate, but the details are a signal of the Republicans’ negotiating position with the White House.

The bill would make able-bodied adults without dependents who receive both federal food assistance and Medicaid benefits subject to work requirements until they are 55 years old, an increase from 49. It also seeks to close a loophole that Republicans have claimed is abused by states, which allows officials to exempt food assistance recipients from work requirements.

Asked if he was open to tougher work requirements for aid programs, Mr. Biden said over the weekend that had voted for such measures as a senator, “but for Medicaid it’s a different story.”

Michael Kikukawa, a White House spokesman, said Mr. Biden “has been clear that he will not accept proposals that take away people’s health coverage.”

“The president has been clear he will not accept policies that push Americans into poverty,” Mr. Kikukawa said.

Conservatives had initially pushed to tighten those work requirements even further, but more mainstream Republicans in competitive districts balked.

Alan Rappeport contributed reporting.