Just days before the Sept. 30 deadline, Democratic Senate Majority Leader Chuck Schumer and Republican Minority Leader Mitch McConnell backed a draft that would keep government institutions open until Nov. 17.
But there was no immediate indication that split House Republicans behind the spat over government funding would adopt the proposal if passed by the Senate.
McConnell said that "shutting down the government for a local budget dispute will not strengthen the political position of any party."
"All it will do is freeze important progress and worry millions of Americans."
The White House joined McConnell in pressuring Congress to pass the Senate package, while President Joe Biden said at a campaign event Tuesday night: "It's time for House Republicans to start doing their job."
Freezing unpaid services and salaries
If no deal is reached by Saturday, hundreds of thousands of federal employees could be temporarily laid off, reducing a wide range of government services.
That could mean interrupted payments to people who rely on subsidies and slowed airport operations, affecting millions of travel plans.
Some employees, such as military personnel and others deemed essential, will continue to work, but will not receive their salaries until the budget is passed.
Biden blamed a small group of "extremist" Republican lawmakers.
In a video message released earlier, he said hardliners in the House of Representatives were "determined to paralyze the government".
Traditional Republicans, meanwhile, refuse to "stand up to extremists within their own party," Biden said.
"So everyone in the United States can now be forced to pay the price."
Aid to Ukraine at Stake
The Senate draft says U.S. economic and military aid to Ukraine, which has been at war for more than a year and a half, could be threatened.
Last week, Ukrainian President Volodymyr Zelensky visited the U.S. Capitol in an attempt to convince the growing number of Republican congressmen wary of aid to Kiev not to abandon his country.
But after Biden asked Congress for $24 billion for Kiev, the short-term proposal released Tuesday allocated only $6.1 billion.
The hardliners in the House of Representatives stressed their refusal to give Kiev any additional money, after the United States provided $110 billion since the war began in February 2022.
House Speaker Kevin McCarthy said Tuesday night: "They prefer Ukraine to Americans."
Instead, he and other Republicans have lobbied for increased funding for border security to combat irregular immigration.
"If they want to focus on Ukraine and not on the southern border, their order of priorities is reversed," he said.
Even if the Senate passes the proposal quickly, it is unlikely that the House of Representatives will be able to act quickly enough to avoid paralysis of public institutions, albeit short.
The prospect that the world's largest economy will not be able to set a government budget comes four months after Washington came dangerously close to a possible default due to political differences.
This could have catastrophic repercussions beyond the U.S. economy.
U.S. stock markets fell more than one percent on Tuesday amid the prospect of a government shutdown.
On Monday, Moody's warned that recent events threaten the U.S. credit rating.