With Congress going into crunch sessions ahead of an 11:59 pm (0359 GMT Tuesday) deadline, a House Republican leader offered a glimmer of hope when he hinted that his party could offer a new plan that might pass muster in the Democratic-held Senate.
"I think the House will get back together in enough time, send another provision not to shut the government down, but to fund it, and it will have a few other options in there for the Senate to look at again," number three House Republican Kevin McCarthy told Fox News Sunday.
Congress must pass a stopgap funding measure before the new fiscal year begins Tuesday or much of the US federal government will close down.
The procedure became dramatically more complicated when Republicans linked the budget legislation to an attempt to thwart President Barack Obama's health care law.
The impasse is already affecting global markets: In Asian trading on Monday oil prices tumbled, markets slumped and the Japanese yen jumped in value while the dollar weakened, mostly due to fears that a budget deal will not be reached before a shutdown.
After the Senate passed a straightforward spending bill on Friday, the House countered after hours of debate Saturday by attaching amendments seeking a one-year delay to so-called Obamacare, as well as repeal of a medical device tax which helps fund the law.
Democratic Senate Majority Leader Harry Reid, who refused to call the chamber into session over the weekend despite the rapidly approaching deadline, warned before the vote that such a measure would be dead on arrival.
The White House also sharply rebuked the move, and warned that the president would veto it even if the Senate did approve it.
On Monday "the Senate will do exactly what we said we would do and reject these measures," Reid spokesman Adam Jentleson said."
"At that point, Republicans will be faced with the same choice they have always faced: put the Senate's clean funding bill on the floor and let it pass with bipartisan votes, or force a Republican government shutdown."
Republican McCarthy hinted that if the Senate rejects the measure as expected, the new House approach would still retain a provision "that I believe the Senate can accept, that will have fundamental changes into Obamacare."
That provision may well be the medical device tax. The Senate voted overwhelmingly to repeal the tax in March, but it was a non-binding budget resolution.
The Senate's number two Democrat, Dick Durbin, appeared open to the possibility.
"I'm willing to look at that, but not with a gun to my head, not with the prospect of shutting down the government," he told the CBS Sunday talk show "Face the Nation."
With Republicans and Democrats in a dangerous game of political chicken, the government was on the verge of ordering hundreds of thousands of federal workers to stay home.
As lawmakers traded blame, the Democratic leadership sounded resigned about a pending shutdown.
Asked if he believed government would shutter on Tuesday, Durbin grimly said "I'm afraid I do."
House Speaker John Boehner has been under intense pressure from his party's most conservative wing, a small band of lawmakers who forced Republican leaders to double down on their anti-Obamacare strategy.
Reid insisted that "the American people will not be extorted by Tea Party anarchists," referring to the ultra-conservative faction of Republicans.
But Boehner warned that if the Senate waits until hours before the deadline to vote, "it would be an act of breathtaking arrogance by the Senate Democratic leadership."