Aides to Senate Majority Leader

Chuck Schumer

(D., N.Y.) didn’t immediately react to the offer, which would buy time but wouldn’t address the underlying dispute over how best to address the debt ceiling longer term.

The offer came as Mr. Schumer is trying for the third time in two weeks to get around Republican opposition in the 50-50 chamber to pass a straightforward increase in the borrowing limit. But following Mr. McConnell’s offer, Democrats delayed their planned vote, which proposed suspending the debt ceiling until the end of next year. Last week, Mr. Schumer tried and failed in two Senate votes to clear the 60-vote hurdle standing in the way of lifting the cap on U.S. debt, currently at about $28.5 trillion.

Republicans say that they don’t want to make it easy for Democrats to raise the debt ceiling because the party has rushed to enact trillions in new spending over GOP opposition. They have urged Democrats to use a process called budget reconciliation, which requires just a simple majority, to pass a debt-limit bill along party lines.

Democrats have rejected that approach, calling it time consuming and unpredictable, and have urged Republicans to let them advance the debt ceiling as regular legislation. Moreover, they say that lifting the debt limit is a shared responsibility, noting that such votes during the Trump administration were bipartisan.

Mr. McConnell, in making his proposal, sought to undercut Democratic claims that using reconciliation procedures would be too time-consuming. “This will moot Democrats’ excuse about the time crunch they created and give the unified Democratic government more than enough time to pass stand-alone debt limit legislation through reconciliation.”

Treasury Secretary

Janet Yellen

has warned that the department is likely to exhaust its cash-conservation measures by Oct. 18 if Congress doesn’t act, saying Wednesday that the country is “staring into a catastrophe.” After that date, she is uncertain whether the Treasury could continue to meet the nation’s commitments, a situation that would force the government to give priority to payments and that she has warned would send the economy into a recession.

Senate Democrats, who met behind closed doors Wednesday morning with White House officials including Susan Rice, haven’t settled on a backup plan if they are unable to pick up 10 Republicans in the 50-50 Senate to end debate and proceed to a simple-majority vote on suspending the debt limit through Dec. 16, 2022.

Democrats have begun discussing whether to curtail the filibuster—the tactic that allows the minority to block legislation by assembling 40 of 100 no votes—to enable a debt-ceiling increase to pass with the votes of 50 senators, with Vice President

Kamala Harris

breaking a tie.

Late Tuesday, President Biden acknowledged that changing Senate rules was a “real possibility.” He has previously said he didn’t want to eliminate the filibuster but could be open to changes.

Still, Sen. Joe Manchin (D., W.Va.), who has opposed ending the filibuster, also opposes making an exception for the debt-ceiling vote, a spokeswoman said Wednesday. That leaves Democrats with few other choices if the two parties remain at odds.

“Work this out; this should not be a crisis,” Mr. Manchin said, calling on Senate leaders to reach a deal. “I’ve been very, very clear where I stand, where I stand on the filibuster. I don’t have to repeat that I think I’ve been very clear. Nothing changed.”

Some lawmakers expressed increasing alarm at the holdup.

“To me this is political stupidity on steroids,” said Sen. Mark Warner (D., Va.). “We are, you know, Thelma and Louise with pedal-to-the-metal, you know, heading towards that cliff, and everybody’s got their heads buried.”

In an August letter, some 46 Republicans wrote that because Democrats had control of both chambers of Congress and the White House, they should use the partisan tactic of budget reconciliation, which allows certain types of legislation tied to spending and taxes to pass on a simple majority vote.

If no Republicans change their positions, that would leave only four potential Republican votes, six short of the 10 that Democrats need. The Republican demand would also force Democrats to assign a dollar value—rather than a date in the future—to the debt ceiling, in an effort at forcing the Democratic Party to take responsibility for debt levels.

A vote to increase the debt limit doesn’t authorize new spending; instead, it essentially allows the Treasury Department to raise money to pay for expenses the government has authorized. Democrats have said that it would be risky to turn to the tool of reconciliation to raise the debt ceiling because of the amount of time it could take to work through the processes associated with the tactic, including two separate sessions of amendment votes in the Senate.

Republicans could reach an agreement with Democrats to speed up the process of using reconciliation to instruct the tax-writing committees to write a separate debt-ceiling bill. But Republicans would have to unanimously agree.

President Biden tried to turn up the heat on Republicans with a meeting Wednesday of business leaders, including those at

Bank of America Corp.

,

Citigroup Inc.,

JPMorgan Chase & Co.,

Nasdaq Inc.,

AARP and the National Association of Realtors, in an attempt to warn against the threat of recession and a downgrading of the U.S. credit rating.

Jane Fraser,

chief executive of Citigroup, described the situation as “playing with fire.” She said a default would cause lasting damage to the U.S.’s reputation and it would hurt consumers and small businesses.

Jamie Dimon,

CEO of JPMorgan Chase, said the debt ceiling should be eliminated altogether to avoid such brinkmanship. He said it was time to show American confidence, “not American incompetence.”

The White House has also highlighted concerns about the effect on average Americans, who could see higher auto- and home-loan interest rates and credit-card payments, the White House said. Tens of millions of seniors who rely on Social Security and Medicare could be affected, officials warned.

“The 2008 financial crisis had ripple effects throughout the global economy that ricocheted back to U.S. shores, causing firms to lay off workers and cut private investment,” the White House Council of Economic Advisers said in a blog post. “A financial crisis driven by a default has the potential to be even worse.”

Mr. Biden said he plans to speak with Mr. McConnell, but as of Wednesday morning, the White House hadn’t detailed any call.

On Wednesday, the Bipartisan Policy Center, a think tank whose senior leadership includes former Republican and Democratic lawmakers, updated its forecast for when the U.S. would no longer be able to meet all its financial obligations, estimating that the so-called X date would likely arrive between Oct. 19 and Nov. 2, a narrower range than the previously expected period of Oct. 15 to Nov. 4.

“Even leading up to Oct. 19, the Treasury Department will find itself with dangerously low cash levels,” said Shai Akabas, the group’s director of economic policy. “An unexpected event during that time frame could escalate into a financial crisis.”

—Andrew Duehren contributed to this article.

Write to Siobhan Hughes at siobhan.hughes@wsj.com and Alex Leary at alex.leary@wsj.com