But some uneasy Senate Republicans are eager to take matters into their own hands.
Moderates like Susan Collins of Maine, conservatives like Rob Portman of Ohio and deal-makers like John McCain of Arizona have quietly begun to reach out to top Senate Democratic leaders to see if they can help break the political logjam. There's no indication the informal talks will lead to a resolution, particularly since the White House and Senate Majority Leader Harry Reid refuse to negotiate before House Republicans agree to raise the debt ceiling and reopen the government with no policy strings attached.
But as the shutdown drags on, the back-channel discussions are a sign of the growing frustration among Senate Republicans that there is no end to the shutdown in sight - and that their House colleagues appear to be at an unbreakable impasse with the White House.
"We have discussions going on, I'm on my way to one right now," said Collins, who spoke with two Senate Democrats Friday morning. "I think that the vast majority of our conference believes the shutdown needs to be brought to an end as soon as possible."
New York Sen. Chuck Schumer, the No. 3 Democratic leader who has helped broker bipartisan deals on immigration and the filibuster "nuclear option" this year, said he's already hearing from Senate Republicans.
"Look, I think there are a good number of Republicans, even very conservative ones that would like to come to an agreement and get the government open again," Schumer said. "There are talks going on, but I'm not going to speak about any details."
Reid added: "I think there are a number of them who have been working very hard to try to get something done. They've come to me."
Portman, the Ohio Republican who was on the short-list to be Mitt Romney's running-mate last year, said he's spoken to seven different Democrats and was "trading ideas back-and-forth." Among his ideas are to push Democrats to lower the spending cap on the continuing resolution from $986 billion to $967 billion, as well as instructions for the House and Senate tax writing committees to produce a tax reform proposal to find $600 billion in mandatory savings outlined in President Barack Obama's budget in order to raise the national debt ceiling by a similar amount.
"There's a growing consensus that we have to do something about spending in the context of the debt limit," Portman said.
While Portman is not demanding proposals to gut Obamacare, like his House GOP counterparts, his calls for spending cuts go much further than Democrats have thus far been willing to tolerate. For instance, Reid said Friday that the "biggest compromise" he's made in 31 years on Capitol Hill was agreeing with House Speaker John Boehner to lower federal funding to $986 billion, the level currently in the Senate's continuing resolution.
And Reid reiterated Friday that he wouldn't compromise further or offer Boehner a concession to end the shutdown.
"This is not a face-saving deal," Reid told reporters. "This is not a date to the prom. This is our country."
Standing in the way of a potential deal is the fact that a number of Senate Republicans are unwilling to undercut Boehner in the high-stakes fight, including Senate Minority Leader Mitch McConnell (R-Ky.), who has stayed mostly out of the spotlight this week but has echoed the speaker's demands for changes in the health care law.
While Boehner and Reid aren't talking, Senate and House Republicans are coordinating closely during the bruising fall fiscal fights. McConnell, Portman and Sen. John Thune (R-S.D.) huddled with House Majority Leader Eric Cantor (R-Va.) and House Budget Chairman Paul Ryan (R-Wis.) on Thursday afternoon, according to a source familiar with the meeting.
Sen. Bob Corker (R-Tenn.), who has been sharply critical of the hardball tactics of tea party Republicans in the government shutdown fight, said it's still up to the House Republicans and Senate Democrats to lead the charge.
"At the end of the day, we're dependent on the House to pass good policy," he said.
Corker was part of a group of Senate Republicans who have previously entertained the idea of a major fiscal deal with the White House. The GOP group repeatedly met with Obama and senior White House officials over fiscal matters this year, though those talks stalled and broke off in August.
Some are renewing those efforts with Senate Democrats.
Sen. Dan Coats (R-Ind.) said Friday he's buttonholed Schumer with some thoughts on how to end the crisis.
"It was very informal, it was talking to somebody who actually has the ear of Harry Reid and the president," Coats said. "Simply saying: This is where Republicans are coming from."
And with public anxiety growing over the shutdown - and Republicans worried they may suffer more of the blame for the gridlock on Capitol Hill - some are clearly ready for the fight to be resolved.
Sen. Ron Johnson (R-Wis.), who joined Senate Republicans in earlier fiscal talks with the White House, said he's ready for the impasse to end.
"This is not a position I want us to be in," Johnson said. "I don't want this brinksmanship - I think the government does enough harm to our economy; we don't need to put on a whole new layer."