By Tom Cohen, Greg Botelho and Leigh Ann Caldwell, CNN
Washington (CNN) -- As talks between House Speaker John Boehner and President Barack Obama over the government shutdown and looming debt limit deadline have hit a brick wall, attention now turns to the Senate where members are working on a separate plan to reopen the government.
The standstill has decreased the possibility that the House would vote to reopen the government this weekend, and while leadership would remain in Washington to continue work, rank-and-file members might return to their districts until Monday afternoon.
In the Senate's court
The Senate is now front and center. The upper chamber defeated a procedural measure Saturday afternoon to extend the debt limit with no strings attached. While the vote was taking place on the Senate floor, the real work is being done behind the scenes.
A bipartisan group of senators have been working on a separate plan. The proposal, led by Republican Sen. Susan Collins of Maine and Democratic Sen. Joe Manchin of West Virginia, would fund the government for six months, extend the debt limit and delay a medical device tax that is part of the Affordable Care Act.
Senate Majority Leader Harry Reid has indicated that he is standing firm on his priorities: passing a "clean" debt limit bill that also reopens the government.
Meanwhile, Tennessee Republican Bob Corker told CNN that Reid and his counterpart, Senate Republican Leader Mitch McConnell, "will continue to talk over the weekend."
"Again, it is very evident that White House is not going to be involved in negotiations, at least at this point they are not, and that the center piece is Reid and McConnell. So I think all of us want to support those efforts and hopefully they bear fruit over the weekend," Corker said.
Corker said that lawmakers continue to talk to each other and that discussions are "good."
Obama and Boehner
Republican leadership told the Republican caucus during a Saturday morning meeting that the president will not accept their proposal to raise the debt limit for six weeks.
The standstill comes after a Friday afternoon phone call between Boehner and Obama, when they decided keep talking.
Rep. Raul Labradorm, R-Idaho, told CNN Saturday that "the President rejected our deal."
Rep. Adam Kinzinger, R-Illinois, said the President is waiting for a better offer.
"It doesn't seem like the White House is serious at all about entering negotiations with us until they see what comes out of the Senate. If they get something out of the Senate that's weaker than our negotiated position, it obviously strengthens their position," Kinzinger said.
The lack of a mechanism in the House proposal to immediately reopen the government -- which has been partially shut down since October 1, prompting the furlough of hundreds of thousands of workers, the closing of national parks and an increase in public anger -- turned off many Senators.
The back story
With many in the Senate, including both Republicans and Democrats, not happy with the House proposal to extend the debt limit but keep the government closed, a bipartisan group started working on their own plan.
Furthermore, some Senate Republicans also see their House colleagues as part of the problem.
There is growing concern being expressed in private meetings that the House GOP plan to extend the debt ceiling for six weeks would come back to haunt the party because it would expire close to the holidays, hurting retailers.
There also are worries about the impact of the partial shutdown dragging on, especially with polls showing the Republican Party getting the brunt of the blame.
Several GOP Senators told CNN that if Boehner fails to make progress on his plan, they'll push the plan backed by moderate Senate Republicans more aggressively.
With today's standstill, that moment might come earlier than expected.
After a meeting Friday at the White House, Collins said Obama "listened carefully" and "said that some of the elements were issues we could work on. But he certainly did not endorse" the proposal.
Sen. Orrin Hatch, R-Utah, said Obama called the medical device tax a "legitimate concern" that can be addressed because it wasn't part of the core Obamacare program.
According to other Senators in the meeting, another Obamacare alteration under discussion would change the definition of a part-time worker in order to protect employees from having their hours cut so businesses can avoid providing coverage.
Some conservatives -- like tea party favorite Sen. Ted Cruz of Texas -- left the meeting upset at Obama for continuing to insist the government must reopen and the debt ceiling must be increased before he would take part in full-fledged negotiations on deficit reduction.
"There was an awful lot of talking, but the president still says he won't negotiate," said Cruz, who spearheaded the attempt last month to add anti-Obamacare amendments to a spending plan needed to prevent the government shutdown.
Source: Obama, Ryan exchange shifts tone
Even as he has continued to firmly repeat his positions, Obama has been reaching across the aisle this week.
One potentially pivotal meeting came Thursday evening, when House GOP leaders went to the White House.
The two sides went around and around for an hour, getting nowhere. Then -- as first reported by CNN's Deirdre Walsh and Dana Bash, citing multiple attendees -- an exchange between Rep. Paul Ryan, R-Florida, and Obama seemed to clear the air.
CNN Chief National Correspondent John King told CNN's "New Day" that Ryan, the chairman of the House Budget Committee, "said something to the effect of, 'Look, we know you don't like our position, we know you probably don't respect our position, but we're the Republican majority.'"
Those comments changed the tone of the meeting, King said. Obama reiterated his opposition to negotiations before the government reopens, then urged GOP leaders to go ask their colleagues what they want done to make that happen.
Republican Rep. Steve Southerland of Florida said the interaction between the two men -- who squared off in the 2012 election when Ryan ran as his party's vice presidential nominee -- "was an important part of the conversation."
"Paul and the President certainly have a past through the last election, and I think there's a great respect between them. And you can't make that up." Southerland said.
The new atmosphere was reflected in both sides' comments after the meeting. The White House called it good and said the parties discussed "potential paths forward."
Rep. Hal Rogers, R-Kentucky, said they were talking "in good faith" about both the debt and reopening the government.
"We're all working together now," said Rep. Pete Sessions, a Texas Republican.
In a video message to a right-wing political summit on Friday, Ryan noted that conservatives can't get everything they want with Democrats holding the White House and a majority of the Senate.
"This President won't agree to everything we need to do," Ryan said, according to excerpts provided by one of his aides. "A budget agreement with this President and this Senate won't solve our problems. But I hope it's a start."
CNN's Deirdre Walsh, Dana Bash, Alan Silverleib, Chelsea J. Carter, Paul Steinhauser, Jim Acosta, Barbara Starr, Ted Barrett, Dan Merica and Brianna Keilar contributed to this report.