WASHINGTON — Senators broke through an impasse late Saturday night over a Republican effort to curtail the powers of the Federal Reserve, clearing away what had been seen as the final hurdle to a deal on a $900 billion stimulus package as lawmakers raced against a Sunday-night deadline to avoid a government shutdown.

With time running out for a compromise, Senator Patrick J. Toomey, Republican of Pennsylvania, agreed to narrow his effort to rein in the central bank, according to three aides familiar with the discussion. All three aides, speaking on the condition of anonymity, noted that the precise language was still being finalized.

Mr. Toomey had sought to bar the Fed and Treasury Department from setting up any loan program similar to those established this year that have helped to keep credit flowing to corporate, municipal and medium-size business borrowers during the recession brought on by the coronavirus pandemic. The agreed-upon alternative, which was still being drafted near midnight on Saturday, aides familiar with the process said, would bar only programs that were more or less exact copycats of the ones newly employed in 2020.

The agreement was a critical breakthrough for lawmakers who have been racing to complete the emergency plan to rush direct payments, unemployment benefits and food and rental assistance to millions of Americans, as well as provide relief to businesses and funds for vaccine distribution. While negotiators were still wrangling over a number of smaller issues, the Federal Reserve language had emerged as the thorniest sticking point to a final agreement.

“If things continue on this path, and nothing gets in the way, we’ll be able to vote tomorrow,” Senator Chuck Schumer, Democrat of New York and the minority leader, told reporters as he left the Capitol shortly before midnight. “House and Senate.”

In an indication that a final deal was close, House leaders notified lawmakers to expect votes as soon as early Sunday afternoon.

President Trump, who has been absent from the frenzied negotiations on the measure, weighed in on Twitter, exhorting Congress to “GET IT DONE” and provide “more money in direct payments.”

The emerging deal would send direct payments of $600 to many Americans and provide enhanced federal jobless payments of $300 per week until early spring. It would also provide hundreds of billions of dollars to prop up small businesses, schools and other institutions struggling amid the pandemic.

With government funding set to lapse Sunday and both chambers hoping to merge the stimulus package with a catchall measure to cover all federal spending for the remainder of the fiscal year, time was dwindling to finalize the recovery plan and rush it to Mr. Trump’s desk.

Without action by Congress, two programs designed to expand and enhance unemployment benefits are set to expire in the coming days, leaving about 12 million Americans without federal support. A number of other benefits are set to expire at the end of the year.

Democrats had said that Mr. Toomey’s proposal, which had been embraced by Republicans, amounted to an attempt to undercut President-elect Joseph R. Biden Jr. and his administration’s ability to continue supporting the country’s economic recovery.

The Fed rolled out a wide-ranging suite of emergency loan programs in response to market disruptions in March and April, and Congress allotted $454 billion to back up those efforts in the first pandemic stimulus law, enacted in March. But Mr. Toomey remained concerned for months that the programs might last past the end of the year. Though Steven Mnuchin, the Treasury secretary, moved to claw back the money in mid-November, Mr. Toomey insisted that the new stimulus measure close off the programs for fear that a new Treasury secretary might try to restart the effort next year, once Mr. Biden takes office.

As initially drafted, Mr. Toomey’s proposal would have prevented the Fed and the Treasury Department from re-establishing emergency lending programs that had been backed by the congressional appropriation, and would have barred the creation of “similar” programs going forward.

While it was not clear exactly what the new language would say, two aides familiar with the agreement said it would prevent the new Treasury secretary from revamping carbon copies of the pandemic lending programs. But under the compromise struck late Saturday, proposed by Mr. Schumer, it would not prevent Fed emergency programs from lending to states or medium-size businesses in the future, so long as the design of the programs was somewhat different, the people said.

Mr. Toomey had been particularly concerned that Democrats might harness the Fed’s powers to funnel cheap money to state and local borrowers. Democrats have been seeking additional avenues to provide relief to those governments.

“This agreement will preserve Fed independence and prevent Democrats from hijacking these programs for political and social policy purposes,” Steve Kelly, a spokesman for Mr. Toomey, said in a statement.

Mr. Toomey and his Republican allies argued that his original proposal merely codified what Congress intended in March when it enacted the original $2.2 trillion pandemic stimulus law, which earmarked funding to support the Fed’s emergency lending programs. But the scope of the language proposed by Mr. Toomey went beyond that, prompting alarm from Democrats, who said they were enlisting prominent figures to weigh in against it.

Ben S. Bernanke, who led the Fed through the 2008 financial crisis, had issued a statement warning that it was “vital that the Federal Reserve’s ability to respond promptly to damaging disruptions in credit markets not be circumscribed.”

Referring to the March stimulus law, Mr. Bernanke added, “The relief act should ensure, at least, that the Federal Reserve’s emergency lending authorities, as they stood before the passage of the CARES Act, remain fully intact and available to respond to future crises.”

Mr. Schumer said that Jerome H. Powell, the current Fed chairman, whom he called “hardly a flaming liberal” was “strongly opposed” to Mr. Toomey’s proposal. The Fed declined to comment on whether Mr. Powell, a Republican who was first nominated as a central bank governor by former President Barack Obama, had discussed the issue with Mr. Schumer in recent days. Mr. Schumer’s office did not respond to a request for more details.

On a private call with House Democrats on Saturday, Speaker Nancy Pelosi of California denounced the proposal, telling lawmakers that “for them to write in there that this cannot happen ever again is just beyond the pale,” according to a person on the call, who disclosed the comments on the condition of anonymity.

“It’s a way for them to say to Joe Biden: ‘We are tying your hands. No matter what comes down the pike, you can’t do this,’” Ms. Pelosi told Democrats.

Mr. Toomey denied that he was seeking to hamstring the Biden administration, and pointed out that he had for months sought to ensure that the Fed’s pandemic programs sunset. But the language he has proposed to attach to the stimulus plan has been more expansive than that.

Republican senators had rallied around Mr. Toomey on Saturday. Senator Mitch McConnell of Kentucky, the majority leader, privately told them during an afternoon call with Mr. Mnuchin that they should stand firmly behind Mr. Toomey.

“Now that Democrats have agreed to a version of Senator Toomey’s important language, we can begin closing out the rest of the package to deliver much-needed relief to families, workers, and businesses,” said Doug Andres, a spokesman for Mr. McConnell.

By early evening, there were indications that Mr. Toomey might be willing to compromise. During the rare Saturday session, he could be seen circulating on the Senate floor, vigorously debating the language with his colleagues. Twice, he was spotted in Mr. Schumer’s leadership suite, including shortly before 8 p.m. on Saturday.

Luke Broadwater and Catie Edmondson contributed reporting from Washington, and Nicholas Fandos from New York.