Senate Majority Leader Senator Harry Reid (D-NV). (Brendan Smialowski/AFP/Getty)

The House bill would have embraced the sequester, deep automatic budget cuts designed to shrink the federal government. The Senate bill would have ended it, restoring billions of dollars for housing, roads and bridges.

This week, congressional Republicans tacitly rejected both approaches to next year’s budget, leaving frustrated lawmakers wondering how they will manage to keep the government open past September, much less resolve a broader conflict over the rising national debt.

As Congress prepared to leave town for a five-week summer break, the prospects for progress on any front in the endless Washington budget war appeared excruciatingly dim. Without an agreement to deal with the sequester and fund federal agencies in fiscal 2014, the government will shut down Oct. 1 — barely three weeks after lawmakers return to town. A few weeks after that, the Treasury will face the risk of default unless Congress can agree to raise the $16.7 trillion federal debt limit.

Leaders of both parties say they want to avoid those outcomes, either of which could seriously damage the sluggish economic recovery. But Republicans have so far refused to open official negotiations with Democrats, either over a budget blueprint for 2014 or over specific spending bills such as the $54 billion transportation and housing measure that Senate Republicans torpedoed Thursday.

“This is so absurd,” Sen. Susan Collins (R-Maine) lamented after every other member of her party voted to block the bill she had painstakingly drafted with Senate Democrats. Collins is among a growing number of Republicans who say the sequester cuts are damaging the government’s ability to perform essential functions such as educating the children of U.S. soldiers and paying private landlords to house the poor.

Ed O'Keefe joins Chris Cillizza and Jackie Kucinich from Capitol Hill to talk about GOP bills aimed at the IRS scandal and Obamacare. (The Washington Post)

If party leaders can’t admit that, Collins said, “I truly don’t know the path forward. I truly don’t. . . . Maybe there will be a grand bargain that will replace all this.”

Just this week, President Obama tried to revive interest in a grand bargain that would pair more tax revenue, long sought by Democrats, with cuts to federal health and retirement benefits long sought by Republicans. During a speech in Tennessee, the president also called for an end to the sequester and sought fresh funding for infrastructure and jobs.

But Republican leaders have so far rejected Obama’s overtures, arguing that ending the sequester — part of a deal to raise the federal debt limit in 2011 — would erode their sole victory in the fight to shrink the size of government. Even approving Collins’s transportation bill would have marked a step backward, said Senate Minority Leader Mitch McConnell (R-Ky.), who twisted the arms of at least four GOP senators who Collins said had pledged to support the measure.

“Regretfully,” McConnell argued in the Senate debate, any vote to ignore the sequester “will be widely viewed throughout the country that we’re walking away from a bipartisan commitment . . . to reduce $2.1 trillion in spending over the next 10 years.”

Senate Budget Committee Chairman Patty Murray (D-Wash.), who worked with Collins to draft the transportation bill, said GOP leaders have maneuvered themselves into a corner. The defeat of the transportation bill came one day after House Republicans refused to support their own version of the measure, which would have set spending on transportation and housing programs at $44 billion next year, nearly 20 percent lower than the Senate bill.

That would have meant cutting grants to state highway departments and slashing community development grants to their lowest level in history, Collins said. After House leaders canceled a vote on the bill, House Appropriations Committee Chairman Harold Rogers (R-Ky.) demanded an end to the sequester, calling it “unrealistic and ill-conceived.”

“The conundrum is that Republicans know sequestration levels don’t work, but they can’t figure out how to open the door to solving this problem,” Murray said. “They have so forcefully fought for the tea party agenda, arguing that there is not one good dime spent at the federal level, that they don’t know how to get off that and move to compromise.”

Obama has opened lines of communication to other GOP lawmakers in an effort to break the deadlock at the top. White House chief of staff Denis McDonough has met several times with House Budget Committee Chairman Paul Ryan (R-Wis.). And he meets regularly with a group of eight Senate Republicans, winnowed from the 24 who had dinner with Obama earlier this year. During a meeting late Thursday at the White House, Obama dropped in for about an hour.

That group — dubbed “the Diners Club” — includes Sen. John McCain (Ariz.), which many Democrats take as a heartening sign. In recent months, McCain has emerged as the leader of a bloc of GOP senators who are fed up with tea-party gridlock and have proved willing to cut bipartisan deals to advance an immigration reform bill and several stalled presidential appointments.

The group also includes Sens. Lindsey O. Graham (S.C.), Kelly Ayotte (N.H.), Bob Corker (Tenn.), Johnny Isakson (Ga.), Daniel Coats (Ind.), Ronald H. Johnson (Wis.) and John Hoeven (N.D.) — “the people we can pass stuff with,” said a senior Democratic aide, speaking on condition of anonymity in order to speak candidly.

But after weeks of talks — including two meetings in the past two days with McDonough — McCain said the group has yet to reach agreement on the most basic issues. Should they try to do a small deal first to replace the sequester and negotiate a broader debt-reduction package later, or do both at the same time? And how should they handle the maze of deadlines looming this fall?

“All those moving parts have really inhibited the ability to seriously address it,” McCain said. “What I fear is that we’re going to again come to the edge of the cliff and all of a sudden there’s going to be these midnight meetings.

“That’s the worst way to legislate,” he said. But “right now, I still don’t see the strategy for the path towards result.” He joked: “It’s always darkest before it’s totally black.”

Several Republican senators not involved in those talks, as well as several senior aides in both parties, said the Diners Club talks were not particularly promising. The group, they said, has failed to agree among themselves on an approach to debt reduction, much less one that would be acceptable to the White House and a broad swath of the GOP caucus.

Graham conceded that cutting a narrower deal between members of the group and the White House was not an option. Any deal that involves overhauling entitlement programs and raising fresh tax revenue is likely to be politically explosive. “It’s got to be 75 [votes in the Senate], or none,” he said.

Several senators have begun talking outside the Diners Club channel, notably Saxby Chambliss (R-Ga.) and Mark R. Warner (D-Va.), members of a bipartisan group that worked for months to forge a debt-reduction deal in 2011. But as lawmakers prepared to head home for their break, hope for a quick resolution was in short supply.

Senior GOP aides in the House said they are counting on Obama to agree to extend the sequester past Oct. 1, keeping the government open and giving lawmakers more time to cut a big deal. Meanwhile, even some members of the Diners Club said they were resigned to spending Christmas in the Capitol for a second straight year.

“Oh, I think that’s inevitable,” Isakson said. “Don’t you?”