WASHINGTON — Days from a deadline, President Biden and Speaker McCarthy are narrowing in on a two-year budget deal aiming to curb federal deficits in exchange for lifting the nation’s debt ceiling and staving off a government default.

The Democratic president and Republican speaker hope to strike a budget compromise this weekend. With Republicans driving for steep cuts, the two sides have been unable to agree to spending levels for 2024 and 2025. Any deal would need to be a political compromise, with support from both Democrats and Republicans to pass the divided Congress.

Yet the budget flow isn’t the only hang-up.

A person familiar with the talks said the two sides are “dug in” on whether or not to agree to Republican demands to impose stiffer work requirements on people who receive government food stamps, cash assistance and health care aid.

Both Messrs. Biden and McCarthy expressed optimism heading into the weekend that the gulf between their positions could be bridged. A two-year deal would raise the debt limit for that time, past the 2024 presidential election.

“We knew this would not be easy,” Mr. McCarthy said as he left the Capitol for the evening Thursday. “It’s hard, but we’re working and we’re going to continue to work until we get this done.”

America could face a default as soon as June 1.

In remarks at the White House, Mr. Biden said, “It’s about competing versions of America.”

“The only way to move forward is with a bipartisan agreement,” Biden said Thursday. “And I believe we’ll come to an agreement that allows us to move forward and protects the hardworking Americans of this country.”

Lawmakers are tentatively not expected back at work until Tuesday, just two days from the early June deadline when the Treasury secretary, Janet Yellen, has said America could start running out of cash to pay its bills and face a federal default.

Mr. Biden will also be away this weekend, departing Friday for the presidential retreat at Camp David, Maryland, and Sunday for his home in Wilmington, Delaware. The Senate is on recess and will be until after Memorial Day.

Meanwhile, Fitch Ratings agency placed the United States’ AAA credit on “ratings watch negative,” warning of a possible downgrade.

Weeks of negotiations between Republicans and the White House have failed to produce a deal — in part because the Biden administration resisted negotiating with 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.

The White House has offered to freeze next year’s 2024 spending at current levels and restrict 2025 spending, but the Republican leader says that’s not enough.

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

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

Pressure is bearing down on Mr. McCarthy from the House’s right flank not to give in to any deal, even if it means blowing past the June 1 deadline.

“Let’s hold the line,” said Representative Chip Roy of Texas, a Freedom Caucus member.

Mr. McCarthy said President Trump told him, “Make sure you get a good agreement.”

Failure to raise the nation’s debt ceiling, now at $31 trillion, to pay America’s already incurred bills would risk a potentially chaotic federal default. Anxious retirees and social service groups are among those already making default contingency plans.

Even if negotiators strike a deal in coming days, McCarthy has promised lawmakers he will abide by the rule to post any bill for 72 hours before voting — now likely Tuesday or even Wednesday. 

The Democratic-held Senate has vowed to move quickly to send the package to Biden’s desk, right before next Thursday’s possible deadline.

In one potential development, Republicans may be easing their demand to boost defense spending, instead offering to keep it at levels the Biden administration proposed, according to another person familiar with the talks.

The teams are also eyeing a proposal to boost energy transmission line development from Senator Hickenlooper. That would facilitate the buildout of an interregional power grid, according to a person familiar with that draft. 

The White House has continued to argue that deficits can be reduced by ending tax breaks for wealthier households and some corporations, but Mr. McCarthy said he told the president as early as their February meeting that raising revenue from tax hikes was off the table.

While Mr. Biden has ruled out, for now, invoking the 14th Amendment to raise the debt limit on his own, Democrats in the House announced they have all signed on to a legislative “discharge” process that would force a debt ceiling vote. They need five Republicans to break with their party and tip the majority to set the plan forward.

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