WASHINGTON — Senator Mitch McConnell, bowing to the immediate threat of a federal default, said Republicans would allow Democrats to raise the debt ceiling into December, but he refused to lift his blockade of a long-term increase in the government’s borrowing limit.

The offer appeared to reflect some nervousness on the part of Republicans in an escalating standoff over the government’s borrowing limit, as a first-ever default on federal debt looms in as few as 12 days. Mr. McConnell has led his party’s refusal to allow Democrats to even take up legislation to lift the cap, instead demanding that they employ a complicated and time-consuming budget maneuver called reconciliation to do so.

But in a statement on Wednesday, he said Republicans would “allow Democrats to use normal procedures to pass an emergency debt limit extension at a fixed dollar amount to cover current spending levels into December.”

The proposal offered a way out of the short-term fiscal crisis facing the nation, and a means for Mr. McConnell to defuse criticism that his party was acting recklessly in blocking Democrats from addressing it. It also confronted Democrats with the prospect of a politically uncomfortable vote that some of them had wanted to avoid, embracing a set dollar amount by which they would raise the debt cap.

“This will moot Democrats’ excuses about the time crunch they created and give the unified Democratic government more than enough time to pass standalone debt limit legislation through reconciliation,” Mr. McConnell said.

Whether or not Democrats agree, the proposal would also do little to resolve the current partisan impasse over the debt limit, merely postponing a default deadline for a matter of weeks.

Mr. McConnell floated the offer as Senate Republicans were set to block a vote on lifting the federal debt ceiling, keeping the government on course to its first-ever loan default, an economic debacle that both sides acknowledge could have devastating consequences.

Senate Democrats need 10 Republicans to side with them to allow them to bring up a House-passed measure that would lift the government’s legal borrowing limit until December 2022, but they appear to be nowhere close. It would be the second time in 10 days that Senate Republicans have filibustered a debt ceiling measure.

And it would leave Democrats with three difficult choices. They could bow to Mr. McConnell’s demand and begin the reconciliation process that would ultimately bypass the blockade; move to change the Senate’s filibuster rules and allow a simple majority vote to raise the debt ceiling; or keep trying and hope that Republicans eventually fold.

Mr. McConnell’s offer was designed to rob Democrats of the argument that they do not have time to capitulate to his demand. The reconciliation process must start in the House with a budget resolution instructing committees to draft legislation to lift the debt ceiling, and the House is not even in session this week. With at least some Republicans promising more dilatory tactics, the process could take awhile, but not until December.

Mr. McConnell may be showing concern about the other two alternatives. Carving out an exception to the filibuster rules, which effectively require 60 votes to move forward with most legislation, has been done before — for judicial and executive branch nominees. But doing so unilaterally to raise the debt ceiling would be a lurch toward ending the filibuster for all policy matters and instituting straight majority rule.

The move has long been resisted by institutionalists and centrists in both parties, including two current Democrats, Senators Joe Manchin III of West Virginia and Kyrsten Sinema of Arizona, although some leading Democrats argue this debt ceiling drama could change minds.

“Nothing changed,” Mr. Manchin told reporters on Wednesday,

Senator Roy Blunt of Missouri, the No. 4 Republican, warned that if Democrats change the filibuster rule, “they’ll permanently change the Senate, permanently change the relationships that still matter in the Senate, and institute the idea that 50 of you plus a vice president of your party can always do whatever you want to do.

“I don’t think that’s healthy for the country,” Mr. Blunt said. “It certainly wouldn’t be healthy for the Senate.”

Praying for Republicans to fold after two failed efforts to break their filibuster is an even riskier play for Democrats, given the stakes. If, for the first time, the U.S. government could not meet its obligations to international lenders, the world economy’s safe-harbor investment would be called into question. Interest rates would probably rise sharply, and global financial institutions would begin searching for new vehicles to store money, where it would not be subject to the whims of partisan politics.

“We’re not asking them to blink; we’re asking them to be the slightest bit reasonable,” Senator Angus King, a centrist independent from Maine, said of Republican leaders. “The political gain of this strikes me as low. The loss to the country strikes me as extraordinarily high.”

Republican intransigence over the borrowing limit forced Democrats last week to strip out a debt-ceiling increase from a must-pass spending bill to avoid a government shutdown. And Mr. McConnell refused to grant consent to allow Democrats to unilaterally move to a vote.

“Democratic leaders haven’t wanted solutions,” Mr. McConnell said on the Senate floor on Wednesday. “They’ve wanted to turn their failure into everybody else’s crisis, playing risky games with our economy, using manufactured drama to bully their own members, indulging petty politics instead of governing.”

Top Democrats have since dropped their insistence that Republicans join them in bipartisan support for raising the statutory cap on the government’s ability to borrow to meet its financial obligations. They, in turn, want Mr. McConnell to honor his demand that Democrats lift the ceiling alone — either by granting consent to move to a vote or by providing 10 Republican votes to break the filibuster.

“We’ve already presented Republicans, numerous opportunities to do what they say they want, including by offering a simple majority vote so Democrats can suspend the debt ceiling on our own as Republicans have asked,” Senator Chuck Schumer of New York, the majority leader, said on Wednesday. “But each time Republicans have chosen obstruction.”

While top Republicans have said most of their members would privately support allowing Democrats to move forward on their own, they have been unwilling to vote publicly that way, and any single senator can object and force one.

Mr. McConnell’s offer indicated he could get all 50 Republicans to agree to allow a vote on a short-term increase in the debt limit.

Democrats also united in opposition to debt ceiling increases under President George W. Bush, but allowed Republicans to do the increases on their own, without filibusters.

“This whole thing drives me crazy because it makes no sense,” Senator Jon Tester, Democrat of Montana, said. “It’s stupid. And yet we’re going down this road.”

But Republican leaders continue to refuse to cooperate on lifting the borrowing limit for a longer period as long as Democrats push forward with their efforts to unilaterally pass a multi-trillion-dollar climate change and social safety net bill, paid for by tax increases aimed at the rich and businesses.

Debt ceiling increases allow the government to pay off debts already incurred, and increasing the borrowing limit would mainly finance repayments for spending and tax cuts from Donald J. Trump’s presidency. But Republicans say Democrats cannot expect their cooperation on one important measure, lifting the debt ceiling, while bulldozing over their objections to their ongoing “tax-and-spend” efforts.

On Wednesday, Mr. McConnell also made Democrats an offer they are sure to refuse: Abandon their effort to pass the expansive domestic agenda on their own and he would extend bipartisan cooperation to raise the debt ceiling.

“If Democrats abandon their efforts to ram through another historically reckless taxing and spending spree that will hurt families and help China, a more traditional bipartisan governing conversation could be possible,” he wrote.