Updated at 2 p.m. ET
The government shutdown is now in its 11th day, but leaders in Washington have turned their focus primarily to the approaching deadline to raise the nation's debt limit.
Senate Republicans headed to the White House Friday morning for a meeting with President Obama, which Senate Minority Leader Mitch McConnell, R-Ky., said could be a "good opportunity to exchange in frank ideas with the president." Alternatively, McConnell continued, if all Mr. Obama wanted to do was "drag us over there to say he won't negotiate, they won't be particularly productive."
After the meeting, Sen. Susan Collins, R-Maine, said it was a good meeting but nothing conclusive came of it.
Mr. Obama met with House Republicans Thursday night for nearly 90 minutes and heard out their offer to pass a no-strings-attached bill to raise the debt limit enough to let the U.S. keep borrowing money for another six weeks.
The president acknowledged the merit of the offer but lobbied them to end the shutdown as part of the deal, CBS News chief White House correspondent Major Garrett reports. Mr. Obama also reiterated that ongoing budget negotiations could and would cover all potential GOP hot button subjects -- but only after the default and shutdown issues were resolved.
Still, House GOP leaders feel strongly that they can't vote to fund the government without getting some kind of face-saving concession, and they believe they convinced the president of that, CBS News Capitol Hill correspondent Nancy Cordes reports. As one aide put it, "It was very clear that this clean debt limit was our coming halfway, and now he needs to have a down payment as well."
Meanwhile, staff members from the White House and House Republican leadership communicated throughout the night over a possible path forward.
A House GOP proposal -- for a short-term debt limit hike that sets up a framework for negotiations on a bigger budget deal -- was delivered to the White House at around 10 or 11 p.m. Thursday night, Rep. Tom Cole, R-Okla., told reporters Friday.
Cole said he expects the White House to offer a response to the plan after Mr. Obama meets with Senate Republicans. If the president's response is positive, Cole thinks an agreement on a short-term spending bill (called a continuing resolution, or CR) would follow. If that were the case, the government could be open by the middle of next week. "It looks to me like the two sides have inched a little closer to each other in the last 24 hours, and I would like them to step closer instead of inch closer, but I think we are moving in the right direction," he said.
Rep. John Fleming, R-La., said House Republicans are looking specifically for an agreement to work on entitlement reform and want to repeal sequester cuts from the military. "I'm not interested in cosmetics here," he said. "I'm not interested in, 'Oh, well we will give you re-assurances that we will talk with you. Now let's open the government up." That's not good enough for me."
That said, Fleming said he'd be willing to give up the demand to delay or dismantle parts of Obamacare as part of the deal -- the GOP demand that caused the shutdown in the first place. "I prefer that, but I understand that not everyone in my conference would necessarily agree to that," Fleming said "That is certainly one of my preferential items -- I think everybody sees how terrible the roll out has been."
On the Senate floor Friday, McConnell maintained that Republicans want to "enact some common sense reforms" before raising the debt limit. Congress must raise the debt limit by Oct. 17 or risk letting the nation default on its loans.
"Here's the key point: nobody wants to default," McConnell said on the Senate floor. "Let's put this hysterical talk of default behind us and instead talk about finding solutions."
Republicans may be willing to vote on a no-strings-attached debt limit bill now, given how poorly they've fared in a number of polls during the shutdown. The latest survey from Wall Street Journal/NBC showed 70 percent of Americans blamed congressional Republicans for putting their political agenda ahead of what's good for the country.
On the Senate floor Friday, Senate Majority Leader Harry Reid, D-Nev., charged that House Speaker John Boehner, R-Ohio, "seemed willing to risk default day after day... to extract extreme political concessions."
Reid said it was encouraging to see his Republican colleagues "come around" to the idea of a no-strings-attached debt limit bill but that a six-week delay was not sufficient to restore economic certainty.
Reid has set up a procedural vote for Saturday on the Senate Democrats' no-strings attached debt limit bill, which would extend U.S. borrowing authority through the end of 2014. It's unclear, however, if Reid has the 60 votes necessary to advance the bill. McConnell said Thursday that Reid's proposal "just won't fly" with Republicans.
Collins said she planned to describe to the president her proposed compromise to address the government shutdown. Her plan would fund the government for six months, repeal the medical device tax (part of Obamacare) and give agencies greater flexibility to deal with across-the-board spending.
Collins said she has had conversations with Senate Republicans and Democrats regarding her plan and hopes it could be used as one potential way to end the shutdown. The House GOP debt limit plan, she added, is "just not enough," since it doesn't address the shutdown.