By Richard Cowan and Katharine Jackson

WASHINGTON Talks on raising the U.S. federal government's $31.4 trillion debt ceiling enter a new phase on Wednesday, after Democratic President Joe Biden and top congressional Republican Kevin McCarthy's first negotiating meeting in three months.

Time is tight to avoid a historic and economically destabilizing default, which the Treasury Department has warned could come as soon as June 1, but some areas of potential compromise emerged after Tuesday's White House meeting.

Biden signaled an openness to Republicans' demand to claw back some unused money for COVID relief, which is less than $80 billion. House of Representatives Speaker McCarthy told reporters that Biden also indicated a willingness to work on speeding up permitting for energy projects, a goal that went unmet in 2022.

The White House has previously supported that idea.

Aides for Biden, McCarthy, top Democratic Senate Majority Leader Chuck Schumer, top Senate Republican Mitch McConnell and top House Democrat Hakeem Jeffries were to begin meeting daily, with another meeting planned on Friday, both sides said.

"Default is not an option," Biden told reporters after Tuesday's meeting. "I told congressional leaders I'm prepared to begin a separate discussion about my budget."

Biden and opposition Republicans have been locked in a standoff for months over the debt ceiling, with Democrats calling for a "clean" increase without conditions to pay debts resulting from spending and tax cuts approved by Congress.

House and Senate Republicans, meanwhile, have said they will not authorize any additional borrowing without an agreement to cut spending.

McCarthy estimated that the two sides had as little as two weeks to reach a deal that could then be passed by Congress.

"I would hope he would be willing to negotiate for the next two weeks so that we can solve this problem," McCarthy said.

Adding to the urgency, Biden is scheduled to leave on May 18 to attend an annual meeting of the leaders of "Group of Seven" major industrialized nations, though he said he would cancel that trip if needed to avoid default.

"If somehow we got down to the wire and we still hadn't resolved this ... I would not go. I would stay until this gets finished," Biden said.

But even if there is significant progress, the House and Senate each have their own, often time-consuming procedures for advancing legislation that opponents of any deal could use to slow things down.

Few countries in the world have debt ceiling laws, and Washington's periodic lifting of the borrowing limit merely allows it to pay for spending Congress has already authorized.

Wall Street executives who have advised the U.S. Treasury's debt operations for the past 25 years warned on Tuesday they were "deeply concerned" about the debt limit impasse, which has markets worried about a U.S. default on payment obligations.

The last time the nation got this close to default was in 2011 - also with a Democratic president and Senate with a Republican-led House.

(Reporting by Richard Cowan and Katharine Jackson; Additional reporting by Moira Warburton; Editing by Scott Malone)