WASHINGTON — The House is poised on Tuesday to approve a new fast-track process for circumventing Republican obstruction and raising the debt ceiling, after leaders in both parties agreed on a measure that could avert the threat of a first-ever federal default.

Speaker Nancy Pelosi of California, who called raising the limit “an imperative,” announced the vote in a letter to Democrats after days of quiet bipartisan talks to resolve the stalemate. Its fate was uncertain in the Senate, but Senator Mitch McConnell of Kentucky, the Republican leader, signaled that it was a solution that he and his colleagues could accept.

The Treasury Department has said that it could breach the statutory limit on its ability to borrow to finance the federal government’s obligations soon after Dec. 15 without congressional action, which would lead to a catastrophic default that could wreak havoc on the U.S. and global economies.

The measure would create a special pathway — to be used only once, before mid-January — for the Senate to raise the debt limit by a specific amount with a simple majority vote, allowing Democrats to steer clear of a filibuster or other procedural hurdles so that Republicans would have no means to block it.

Democrats are likely to choose a number that would keep the government from reaching the new debt ceiling before the midterm elections next year, but senior aides familiar with the talks did not immediately provide an amount.

The unusual maneuver is necessary because Republicans have refused to drop their blockade against legislation to provide a long-term increase in the debt limit, going beyond merely opposing the increase and refusing to clear the procedural hurdles so Democrats could do so unilaterally.

The new bill would effectively strip them of that opportunity for a brief time, a solution that Mr. McConnell said would break the logjam.

“I believe we’ve reached here a solution to the debt ceiling issue that’s consistent with Republican views of raising the debt ceiling for this amount at this particular time and allows the Democrats to proudly own it,” he said at a news conference on Tuesday.

The proposal is wrapped into legislation that would postpone scheduled cuts to Medicare, farm aid and other mandatory spending programs that were set to kick in next year. Once that bill becomes law, Senator Chuck Schumer of New York, the majority leader, would introduce separate legislation raising the debt limit. That is expected to pass with only Democratic votes in the 50-50 Senate, where Vice President Kamala Harris is empowered to break ties.

Mr. McConnell and 10 Republicans agreed in October to allow the Senate to take up a short-term increase to the debt limit, which ultimately passed with Democratic votes. But some of those senators warned they would not acquiesce again, and Mr. McConnell and Mr. Schumer began quietly discussing alternatives.

“I’m confident that this particular procedure, coupled with the avoidance of Medicare cuts, will achieve enough Republican support to clear the 60-vote threshold,” Mr. McConnell said, predicting a Thursday vote for the bill in the Senate.

That would require 10 Republicans to join Democrats in advancing the bill, a prospect that Mr. McConnell discussed at lunch with members of his party on Tuesday afternoon.

Some lawmakers said they would be open to supporting the legislative gymnastics in the interest of foisting political responsibility for raising the debt ceiling on Democrats.

“To have Democrats raise the debt ceiling and be held politically accountable for racking up more debt is my goal, and this helps us accomplish that,” Senator John Cornyn, Republican of Texas, told reporters before the lunch.

“To me, that’s the most important part: that the voters in 2022 see that the people who recklessly spent money that our future generation is going to have to pay back didn’t care,” he added.

The legislation would also postpone until 2023 mandatory cuts to a range of federal spending programs, including farm aid, community block grants and a 4 percent reduction in Medicare payments to doctors and hospitals.

It would extend a temporary pay increase in Medicare that was passed earlier in the coronavirus pandemic. Instead of facing a separate 2 percent cut in January, doctors and hospitals would keep their current pay rates until April, with the cut phasing in slowly afterward.