Nearly two hours after the end of what appeared to be a fruitless meeting between President Biden and Republican leaders this week over raising the federal debt limit, Mr. Biden finally offered a hint as to how the government might avoid a catastrophic default.

Actually, it was two hints, contained in an impromptu news conference in the Roosevelt Room. For the first time, Mr. Biden signaled he was open to clawing back some unspent stimulus money included in the economic rescue bill he signed in 2021 during the pandemic. That suggested a potential starting ground for compromise between the president and Republican lawmakers, who have refused to raise the nation’s borrowing limit without deep spending cuts, and who have pushed to rescind some stimulus funds.

Republicans estimate those clawbacks could save the government between $50 billion and $70 billion, a relatively small amount in the context of their $4.8 trillion proposal to cut spending and eliminate clean-energy tax breaks in exchange for raising the debt limit. Asked about the idea on Tuesday, Mr. Biden did not rule it out.

“We don’t need it all,” Mr. Biden said, referring to the unspent coronavirus funds. “But the question is what obligations were there made, commitments made, money not disbursed, etc.? I have to look, take a hard look at it. It’s on the table.”

The second clue was not about possible compromise — it was the opposite. The president acknowledged that he is considering what would effectively be a constitutional challenge to the very existence of the debt limit. It is a unilateral path that Mr. Biden conceded could face legal hurdles. But his comments suggested that if Congress could not find a deal to raise the debt limit on terms acceptable to Mr. Biden — and before the nation runs out of cash to pay its bills — the president might be prepared to try to avoid default on his own.

“I am considering the 14th Amendment,” Mr. Biden said, referring to a clause in the Constitution that stipulates that “the validity of the public debt” issued by the U.S. government “shall not be questioned.” Some constitutional scholars contend that clause requires the government to continue issuing new debt to pay bondholders, effectively overriding the nation’s statutory borrowing limit, which is controlled by Congress.

Still, Mr. Biden noted, “the problem is it would have to be litigated.”

Neither route is anything close to a safe bet to avoid what economists on Wall Street and in the White House say would be a job-obliterating financial crisis and recession in the event of a prolonged default. The Treasury Department has estimated that it could exhaust its ability to pay all of its bills on time — including payments to America’s lenders — as soon as June 1 if the debt limit is not raised.

Together, though, those options could form the basis for Mr. Biden telling reporters that “I’m absolutely certain” the default threat could be defused in time.

Republicans have forced debt-limit showdowns before, under Mr. Biden and former President Barack Obama. But current and former administration officials, business lobbyists in Washington and even progressive economists say this standoff is different — and could carry significantly higher risks of tipping the nation into default.

Speaker Kevin McCarthy commands only a narrow majority in the House. It includes a large group of spending hawks who have pushed for confrontation over the debt limit. They insisted he tie spending cuts, a rollback of the president’s climate agenda and new work requirements for recipients of government aid to a bill the House passed late last month to raise the debt ceiling, which is capped at $31.4 trillion.

Mr. Biden negotiated a last-minute deal to raise the limit and avoid default in 2011, as vice president. But as president, he has said he will not bargain over the basic responsibility of Congress to allow the government to continue borrowing money to pay its bills — which is needed because lawmakers have spent more than the government receives in tax revenue for more than two decades.

Those dynamics could make it significantly more difficult than it was in 2011 for the country to escape default. Emerging from the White House meeting on Tuesday, Mr. McCarthy immediately sought to blame Mr. Biden and Senate Democrats for the threat of default.

“I’ve done everything in my power” to avoid default, he said. “We have passed a bill that raised the debt limit. Now, I haven’t seen that in the Senate. So I don’t know.”

Mike Konczal, a director of the liberal Roosevelt Institute, said this week that he worried that investors and others could conclude from history “that a solution will show up at the last minute.”

“That seems much less likely now,” he said.

Still, business groups and some current and former administration officials have publicly and privately described what they believe could be the possible contours of a compromise to avoid default. It would most likely center on an agreement to limit federal discretionary spending for at least two years. It could perhaps include other areas of mutual interest to Mr. Biden and Republicans — like streamlining the permitting process to develop wind, oil, gas, solar and other energy projects.

Such a deal could be aided by a plunge in the stock market as a possible default approaches and investors panic, Democrats and Republicans say.

A spending compromise would be easy enough for congressional staff to draft in a matter of hours. Mr. Biden and Senator Chuck Schumer of New York, the majority leader, said on Tuesday that staff members from both parties would meet for two days to discuss spending issues. “We have to sit down and figure it out,” Mr. Schumer told reporters Tuesday. “But taking default off the table would be the best thing to do right now.”

Until Mr. Biden expressed openness to clawing back some coronavirus spending, however, it was unclear whether any urgent fiscal talks could coexist with his insistence that he will not bargain over the debt limit. Now, it seems more likely that Mr. Biden and his team will be willing to discuss some Republican spending proposals on an expedited timeline, whether they call it a negotiation or not.

It is also clearer that Mr. Biden might be willing to act on his own if those discussions do not bear fruit, by invoking the 14th Amendment and continuing to borrow money to pay the government’s bills. “I don’t think that solves our problem,” he said, unless a court rules in the administration’s favor on the issue.

But Mr. Biden said the move was under consideration regardless, and he punctuated that thought at the end of his remarks on a potential default. “I will do everything in my power to avoid it,” he said.

Catie Edmondson contributed reporting.