WASHINGTON — Work requirements for federal food aid recipients have emerged as a final sticking point in negotiations over the looming debt crisis, even as President Biden said Friday that a deal is “very close.”

Mr. Biden’s optimism came as the deadline for a default was pushed back to June 5 and seemed likely to drag negotiations between the White House and Republicans over raising the debt ceiling into another frustrating week. 

Both sides have suggested one of the main holdups is the debate over boosting work requirements for recipients of food stamps and other federal aid programs, a longtime Republican goal Democrats have strenuously opposed.

Even as they came closer to a framework on spending, each side seemed dug in on the work requirements. A White House spokesman, Andrew Bates, called the GOP proposals “cruel and senseless” and said Mr. Biden and Democrats would stand against them.

Congressman Garret Graves, one of Speaker McCarthy’s negotiators, was blunt when asked if Republicans might relent on the issue: “Hell no, not a chance,” he said.

The later “ X-date,” laid out in a letter from Secretary Yellen, set the risk of a devastating default four days beyond an earlier estimate. Still, Americans and the world uneasily watched the negotiating brinkmanship that could throw the American economy into chaos and sap world confidence in the nation’s leadership.

Yet Mr. Biden was upbeat as he left for the Memorial Day weekend at Camp David, declaring, “It’s very close, and I’m optimistic.”

With Republicans at the Capitol talking with Mr. Biden’s team at the White House, the president said: “There’s a negotiation going on. I’m hopeful we’ll know by tonight whether we’re going to be able to have a deal.” 

Yet a deal had not come together when Mr. McCarthy left the Capitol Friday evening.

In a blunt warning, Ms. Yellen said failure to act by the new date would “cause severe hardship to American families, harm our global leadership position and raise questions about our ability to defend our national security interests.”

Anxious retirees and others were already making contingency plans for missed checks, with the next Social Security payments due next week.

Messrs. Biden and McCarthy have seemed to be narrowing on a two-year budget-slashing deal that would also extend the debt limit into 2025 past the next presidential election.

Yet talks over the proposed work requirements for recipients of Medicaid, food stamps and other aid programs seemed at a standstill Friday afternoon.

Mr. Biden has said the Medicaid work requirements would be a nonstarter. But he initially seemed open to possible changes on food stamps, now known as the Supplemental Nutrition Assistance Program, or SNAP.

The Republican proposal would save $11 billion over 10 years by raising the maximum age for existing standards that require able-bodied adults who do not live with dependents to work or attend training programs. 

While current law applies those standards to recipients under the age of 50, the House bill would raise the age to include adults 55 and under. The GOP proposal would also decrease the number of exemptions that states can grant to some recipients subject to those requirements.

Mr. Biden’s position on the food stamps work requirements appeared to have hardened by Friday, when his spokesman, Mr. Bates, said House Republicans are threatening to trigger an unprecedented recession “unless they can take food out of the mouths of hungry Americans.”

Any deal would need to be a political compromise, with support from both Democrats and Republicans to pass the divided Congress. Failure to lift the borrowing limit, now $31 trillion, to pay the nation’s incurred bills, would send shockwaves through the global economy.

But many of the hard-right Trump-aligned Republicans in Congress have long been skeptical of Treasury’s projections, and they are pressing Mr. McCarthy to hold out.

As talks pushed into another late night, one of the negotiators, Congressman Patrick McHenry, called Mr. Biden’s comments “a hopeful sign.” But he also cautioned that there are still “sticky points” impeding a final agreement.

While the contours of the deal have been taking shape to cut spending for 2024 and impose a 1 percent cap on spending growth for 2025, the two sides remain stuck on various provisions.

Weeks of negotiations between Republicans and the White House have failed to produce a deal — in part because the Biden administration resisted negotiating with Mr. McCarthy over the debt limit, arguing that the country’s full faith and credit should not be used as leverage to extract other partisan priorities.

“We have to spend less than we spent last year. That is the starting point,” said McCarthy.

One idea is to set the topline budget numbers but then add a “snap-back” provision to enforce cuts if Congress is unable during its annual appropriations process to meet the new goals.

Lawmakers are all but certain to claw back some $30 billion in unspent Covid funds now that the pandemic emergency has officially been lifted.

Mr. McCarthy has promised lawmakers he will abide by the rule to post any bill for 72 hours before voting. The Democratic-held Senate has vowed to move quickly to send the package to Mr. Biden’s desk.