With Washington barreling toward a government shutdown, a deadlocked Congress entered the final weekend of the fiscal year with no clear ideas of how to avoid furloughs for more than 800,000 federal workers. Millions more could be left without paychecks.

The Senate on Friday approved a stopgap government funding bill and promptly departed, leaving all of the pressure to find a solution on House Republican leaders.

President Obama weighed in, sternly lecturing GOP leaders that the easiest path forward would be to approve the Senate’s bill, which includes money for the implementation of the Affordable Care Act, the president’s prized legislation achievement, which he signed into law in 2010. But a far-right bloc of House and Senate Republicans banded together to leave House Speaker John A. Boehner (R-Ohio) virtually powerless to act.

“My message to Congress is this: Do not shut down the government. Do not shut down the economy. Pass a budget on time,” Obama said in the White House press briefing room.

Boehner’s leadership team offered no public comment and remained out of sight most of Friday, hunkering down for another weekend on the brink. For Boehner, this is the latest in a series of unstable moments that have become the hallmark of his three-year run as speaker.

With a stroke-of-midnight deadline Monday, Senate Majority Leader Harry M. Reid (D-Nev.) said Democrats would reject any conservative add-ons that Boehner might attach to the funding bill. That would further delay passage, and given the staunch opposition from Sen. Ted Cruz (R-Tex.), who has suggested that he will not help move the process along, the slow-moving Senate would require up to a week to approve something even if Reid were amenable to the changes. That sets the stage for a shutdown Tuesday.

“We’ve passed the only bill that can avert a government shutdown Monday night. I said this on the floor, I say it again: This is it, time is gone,” Reid said Friday after the midday passage of the funding bill on a party-line vote.

Before that final roll call, Cruz’s attempt to filibuster the legislation was throttled in a bipartisan 79-to-19 vote, but the first-year senator drew support from nearly half the rank-and-file Republicans in defiance of Senate Minority Leader Mitch McConnell (R-Ky.).

Cruz confirmed reports that he has been huddling with House conservatives to help plot their strategy to force Boehner’s hand on Obamacare. “I am confident if the House listens to the people, as it did last week, that it will continue to step forward and respond to the suffering that is coming from Obamacare,” Cruz told reporters Friday, saying he has had “numerous conversations” with House Republicans.

Those Republicans upended a strategy crafted by Boehner and House Majority Leader Eric Cantor (R-Va.) to first advance legislation related to the federal borrowing limit, including more demands to delay Obamacare, then allow government funding to be approved.

That plan required the GOP leaders to draw all votes from their side of the aisle — 217 of the 232 Republicans — and instead the Cruz-backed contingent hold more than enough votes to sabotage any moves by Boehner and Cantor. Those House Republicans offered their version late Friday of what they want attached to the funding resolution and sent back to the Senate: an amendment delaying until 2015 implementation of all the health law’s taxes, mandates and benefits as well as its provisions aimed at squeezing savings from Medicare.

“A simple and reasonable way to ensure fairness for all is to provide every American the same one-year Obamacare delay that President Obama provided for businesses and others,” Rep. Tom Graves (R-Ga.), the bill’s author, said in a written statement.

He has more than 60 co-sponsors.

While the health-care law has had some provisions delayed amid a wobbly rollout, Obama and Democrats oppose any effort to strip funding or delay implementation of the law as it begins a critical new period next week. The president warned that demands to delay Obamacare were even more reckless in connection with the raising debt limit, because the Treasury will run out of maneuvers to continue borrowing Oct. 17 and will head toward a first-of-its-kind default on the nearly $17 trillion debt. Economists have warned that a default would send a shock through global financial markets and would jolt interest rates.

“I don’t know how I can be more clear about this: Nobody gets to threaten the full faith and credit of the United States just to extract political concessions,” Obama said Friday.

Meanwhile, House leaders delayed consideration of their initial proposal to raise the federal debt limit until at least next week.

It was unclear Friday whether the debt-limit bill would require additional surgery, senior GOP aides said, since most of those who objected to the measure were concerned primarily about timing. However, a separate bloc of lawmakers complained that the bill — a grab bag of conservative agenda items ranging from tax reform to the rollback of environmental regulations — would do too little to cut spending. As written, the measure contained only around $200 billion in spending cuts over the next decade. Meanwhile it would suspend the debt limit through Dec. 5, 2014, permitting the Treasury Department to borrow an additional $1 trillion.

The bill has no hope of passing the Democratic-controlled Senate.

After a few noncontroversial votes naming federal buildings, the House adjourned Friday morning amid deep uncertainty about its next steps. Boehner and Cantor have called a noon Saturday caucus meeting in the Capitol basement to try to forge ahead.

For the moment, GOP leaders have given no indication they were willing to simply approve the Senate legislation. Such a move, some Republicans privately fear, could lead to a collapse of support among GOP lawmakers and result in the legislation passing largely on the strength of Democratic votes. That would leave Boehner, already the weakest speaker of the modern political era, even more politically wounded heading into the debt ceiling talks.

Several Republicans said Friday that they favor a “stick” approach — an amendment so distasteful to Democrats that they might feel compelled to return to the negotiating table. Others favor a “carrot” approach, attaching an item Democrats would find hard to refuse — including possibly delaying sequestration cuts for a year in exchange for delaying implementation of Obamacare for a year. They did not detail the specifics of either approach.

However, with Graves holding potentially several dozen votes, no Republican could offer a sound explanation for how they would avert a shutdown next week.

Before the Senate votes, Reid denounced as “anarchists” the Cruz-led Republicans who he said were driving the country toward economic devastation.

“Today the Republican Party has been infected by a small destructive faction,” Reid said. “These extremists are more interested in putting on a show, as one Republican colleague put it, than legislating.”

The situation is in such flux that some of the most strident conservatives cast votes to filibuster the government funding bill — effectively endorsing shutting down the government — and yet immediately after warned it would not succeed in hindering the health law.

“Obamacare will continue. America’s going to have to judge whether it’s a good thing or bad thing. I still think Obamacare is going to be bad for part-time workers, for workers who may lose their insurance. I think it’s bad for the country,” said Sen. Rand Paul (R-Ky.), a leading contender for his party’s 2016 presidential nomination.

He suggested that the fight against Obamacare had been lost for now and that the GOP should move on to other issues.

One veteran of the mid-1990s shutdowns, which also pitted a Democratic president against a Republican speaker, warned a temporary shutdown was increasingly likely.

“It depends if wisdom trumps energy. It hasn’t thus far, has it?” Sen. Tom Coburn (R-Okla.) said, a dig at those who want to continue the campaign against Obamacare.

Coburn, a freshman House member in the 1990s shutdowns, said it wouldn’t matter much until Oct. 15. That’s when the first paychecks for service members — including those on the front lines of Afghanistan — would not go out.

“When you start getting into military pay, that’s serious. When the people defending this country can’t pay their house payments, things they need to do. . . . We’ll fold like hot cakes if they shut down. Republicans will,” Coburn predicted.

Rosalind S. Helderman, William Branigin and Jackie Kucinich contributed to this report.