Republicans and Democrats appear to be hardening their positions as both sides hunker down for what could be a prolonged fight, with a US government shutdown in its second day.
Democrats demanded that US President Donald Trump negotiate on immigration issues as part of any agreement to resume government funding.
They accused him of reneging on an earlier accord to protect "Dreamers," illegal immigrants brought to the United States as children, from deportation.
"I hope it is just a matter of hours or days. But we need to have a substantive answer, and the only person who can lead us to that is President Trump. This is his shutdown," Dick Durbin, the second ranking Democrat in the Senate, said on the CBS "Face the Nation" program on Sunday.
Republicans were just as adamant, saying they would not negotiate immigration or other issues as long as all but essential government services remain shuttered.
Speaking to US troops at a military base in the Middle East, Vice President Mike Pence said, "We're not going to reopen negotiations on illegal immigration until they reopen the government and give you, our soldiers and your families, the benefits and wages you've earned."
A bipartisan group of senators met on Sunday in a Senate office building, searching for ways out of the crisis.
Moderate Republican Senator Susan Collins said a group of as many as 22 senators were discussing alternatives, though the details were "in flux."
After funding for federal agencies ran out at midnight on Friday, many US government employees were told to stay home or in some cases work without pay until new funding is approved. The shutdown is the first since a 16-day closure in October 2013.
Trump said on Sunday that if the stalemate continued, Republicans should change Senate rules so a measure could be passed to fund the government.
Current Senate rules require a super-majority of three-fifths of the chamber, usually 60 out of 100, for legislation to clear procedural hurdles and pass.
"If stalemate continues, Republicans should go to 51 percent (Nuclear Option) and vote on real, long term budget," Trump said on Twitter.
But Senate Republican Leader Mitch McConnell, from Trump's own party, rejected the idea.
Republicans hold a slim 51-49 majority in the Senate.
The Senate will vote early on Monday on whether to advance a measure to fund the government through February 8, unless Democrats agree to hold it sooner, McConnell said on Saturday.
The level of support for the bill was uncertain, but given Democratic leaders' public statements, it seemed unlikely the measure would receive the 60 votes required to advance.
In a Senate floor speech on Sunday, McConnell accused Senate Democratic leader Chuck Schumer of imperiling children's health care, military training, veterans' care and other programs.
Schumer and his colleagues accused Trump of being an unreliable negotiating partner, saying the two sides came close to a deal on immigration several times, only to have Trump back out under pressure from anti-immigration conservatives.
