“No one can guarantee there won’t be a default, if for no other reason than the clock is ticking down here pretty quickly,” said G. William Hoagland, a longtime Republican budget guru on Capitol Hill who is now a senior vice president at the Bipartisan Policy Center. “We are on thin ice in a big way.”

Negotiators got some breathing room Friday afternoon with the Treasury secretary’s announcement that the default deadline had moved four days later, to June 5. But Congress will still be hard-pressed to act by then, and the brief extension might even be counterproductive, sapping some urgency to seal a deal.

“We’re within the window of being able to perform this, and we have to come to some really tough terms in these closing hours,” said Representative Patrick T. McHenry, Republican of North Carolina and a lead negotiator for Mr. McCarthy. “We’re going back on final, important matters, and it’s just not resolved.”

Since the beginning of the impasse, Mr. Biden and congressional leaders have sought to tamp down concern that a default would occur, essentially saying that it was unthinkable because Congress has narrowly avoided default before. After one of the high-level meetings at the White House, Senator Chuck Schumer, the New York Democrat and majority leader, cheered the fact that all four leaders had said default was off the table.