Congress has until midnight Saturday to pass a new budget before thousands of federal workers are placed on unpaid leave.
The Senate moved ahead with the proposed interim funding agreement, which House Republicans opposed.
The two houses of Congress are controlled by different parties.
The Republican-led House will vote on its own short-term spending bill on Friday. Their plan would extend funding through the end of October, but it calls for deep spending cuts that would make it unacceptable to Democrats, who run the Senate.
The conflicting House and Senate funding plans have raised the possibility of a shutdown and highlighted deep divisions within the Republican Party, with some lawmakers expressing exasperation with conservatives hardliners, who have so far blocked a financial deal.
For example, California Republican Mike Garcia told Reuters he felt “disappointed” as the shutdown approached.
“We are not in a good position for negotiations with the Senate,” he said.
The Senate overwhelmingly passed a procedural vote on a short-term funding bill that had bipartisan support in the House. The bill would avoid a government shutdown until November 17, giving Congress more time to reach a long-term budget deal.
But for the bill to become law, it needs to pass the House, where at least nine hard-line members of Mr. McCarthy’s narrow Republican majority have refused to support the temporary measure.
That group has repeatedly threatened to remove Mr. McCarthy as House Speaker if he relied on Democratic votes to overcome their opposition and pass the funding bill without them.
Mr. McCarthy previously said he had no plans to bring the Senate bill to the House.
“I think the President is making a choice between the presidency and American interests,” President Joe Biden said Wednesday. “This is not an unsolvable puzzle,” Democratic Senate Majority Leader Chuck Schumer said. “President McCarthy must stop letting Maga radicals dictate his decisions.”
Mr. McCarthy did not commit to bringing the Senate short-term funding bill to the House, but said a shutdown could be avoided if Democrats better addressed border security in the bill.
“We’re trying to see if we can get border provisions into the current Senate bill to really improve things,” he told reporters on Thursday.
“This morning I talked to a number of Democratic senators who are more sympathetic to what we want to do. They want to do something about the border issue.”
He later added that he had spoken with Mr. Biden about establishing a national commission to study the national debt and find ways to reduce it.
Biden’s top aide, chief of staff Jeff Zients, told NPR News that the White House is preparing for the possibility of a last-minute deal on Saturday. But he added that Mr. Biden has no plans to meet privately with Mr. McCarthy.
“There is no need for a meeting at this time. The meeting that needs to happen is in the House of Representatives, where House Republicans meet and fund the government.”
In recent months, Mr. McCarthy has noticed a growing ideological divide between moderates and the right in his caucus.
This right wing has only a handful of members, but in a chamber where Republicans control just nine seats, they wield disproportionate power in the debates.
Fiercely opposed to anything resembling the status quo in Washington, they demand the party respect spending cuts and what they call conservative priorities. That means Mr. McCarthy may have to turn to the opposition party to bail him out and support a short-term spending bill.
Working with Democrats is virtually guaranteed to trigger the so-called nullification movement, the first step in forcing a vote to remove him as president.
Economists also warned that a government shutdown could harm the US economy, with Moody’s warning that it could harm the US credit rating and “underscore the weakness lack of US institutional strength and governance”.