WASHINGTON (CNN) -- President Barack Obama and House Speaker John Boehner agreed in a phone call Friday that talks should continue on ending the partial government shutdown and avoiding a possible default, but the President "has some concerns" with the latest House GOP proposal considered an opening gambit in budget brinksmanship, White House spokesman Jay Carney said.
A day of talks at the White House and in Congress raised various ideas for elements of an agreement -- including some changes to Obama's signature health care reforms and a way to raise revenue to prevent forced cuts in government spending -- but little tangible progress toward an immediate breakthrough acceptable to Obama, the GOP-led House and the Democratic-led Senate.
Raising the debt ceiling and reopening the government is "the very least that Congress could do," Carney told reporters.
He confirmed that Obama and Boehner spoke earlier in the day and agreed that everyone should "keep talking."
Carney reiterated that Obama opposes negotiations on broader budget and deficit reduction issues during a shutdown and under the threat of a possible first-ever U.S. default.
The House proposal on Friday followed a White House meeting the day before between the President and the House GOP leadership that participants described as positive in tone.
It offers to increase the federal borrowing limit, which the Obama administration says must occur by October 17 to avoid a potential default that economists warn would harm the economy.
A House vote on the GOP plan could come as soon as this weekend.
"Let's do a temporary debt ceiling, then let's concentrate on the continuing resolution," Rep. Peter Roskam of Illinois, a member of the House GOP leadership, told reporters on Capitol Hill.
Boehner outlined a similar proposal on Thursday, prior to his leadership team's meeting with Obama, and it was unclear if the GOP offer to the White House on Friday contained any substantive changes.
On Thursday, Obama and top Democrats made clear they rejected the initial House GOP approach, but Carney indicated the President would sign a "clean" measure to raise the debt ceiling that contained no partisan policy provisions.
Carney also said Thursday that Obama a debt ceiling increase and the government reopened before launching broader budget and deficit reduction negotiations with Republicans.
Meanwhile, Obama met Friday with Senate Republicans as part of new outreach to Congress as the shutdown continued and the deadline for a possible default loomed closer. Carney said Obama felt that meeting was "constructive."
In the Senate, a separate proposal under consideration would raise the debt ceiling and include a spending provision to reopen the government.
It also includes a Republican priority also backed by many Democrats to end a tax on medical devices created by Obama's signature health care reforms, known as Obamacare.
Sen. Susan Collins of Maine, a moderate Republican pushing the Senate plan, called Friday's meeting a good but "inconclusive" exchange with the President.
Obama "listened carefully" and "said that some of the elements were issues we could work on, but he certainly did not endorse" the Senate proposal, Collins said.
Conservative Sen. Orrin Hatch of 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, one of the elements under discussion would change the definition of a part-time worker under Obamacare in order to protect employees having their hours cut so that businesses can avoid a requirement to provide health coverage.
Some conservatives, however, said Obama refused to budge from his insistence that 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 Sen. Ted Cruz of Texas, a tea party conservative who spearheaded the attempt last month to add anti-Obamacare amendments to a spending plan needed to prevent the government shutdown.
Obama also met with Democratic Party caucuses from the House and Senate this week, but the separate talks with the GOP caucuses were the focus of possible progress in the stalemate that has stoked fear among investors and angered the American public.
At the Thursday night meeting with House GOP leaders at the White House, the two sides went around and around for an hour, getting nowhere.
Then came a sudden change that might ultimately help break the political impasse causing the continuing shutdown and the looming threat of a further crisis when the nation bumps up against its self-imposed borrowing limit.
As first reported by CNN's Walsh and Dana Bash, an exchange between GOP Rep. Paul Ryan and Obama seemed to clear the air, with each side acknowledging the validity of the other.
Ryan, the Republican chairman of the House Budget Committee, conceded his side wouldn't get all it wanted while Obama said, tell me what you need to make something happen, Bash and Walsh reported after the Thursday night meeting.
CNN Chief National Correspondent John King told CNN's "New Day" on Friday that Ryan "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.' "
"You're stuck with us for a while, at least through the next election season, so we need to learn to have a conversation with each other," King paraphrased Ryan as saying.
"And at that point, both Democrats and Republicans say, the tone of the meeting changed," King added. "The president said, 'Listen, I'm not going to negotiate with you until you reopen the government, but go to your members, find out what you need to do to get that part done and let's try to make some progress.'"
CNN's reporting on the meeting is based on accounts from multiple sources who attended.
Republican Rep. Steve Southerland of Florida essentially confirmed the CNN account on Thursday night.
"Paul and the President certainly have a pass through the last election and I think there's a great respect between them. And you can't make that up." Southerland said, adding that "the communication between Paul and the President, I think, was an important part of the conversation."
The meeting appeared to begin breaking the partisan logjam that has led to the shutdown, leaving hundreds of thousands of federal employees without work and caused countless ripple effects, from lost tourist revenue around national parks to a threat to the Alaskan crab fishing season.
"We're all working together now," Rep. Pete Sessions, a Texas Republican, said after the meeting, while House Majority Leader Eric Cantor called it "very useful."
Rep. Hal Rogers, a Kentucky Republican, said both sides are talking "in good faith" about not just the debt ceiling, but also what it will take to restart the government.
"There was not a timeline set," Rogers said. "But we want to move quickly."
The Obama administration described the meeting as "good," saying the President listened to Republican proposals and the two sides discussed "potential paths forward."
Senators are involved in their own talks to come up with a package that would reopen the government and lower the debt ceiling, while ending a tax on medical devices under Obamacare and setting up broader negotiations on deficit reduction.
In a video message to a conservative political summit Friday, Ryan warned the right-wing gathering that they 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," said the message from Ryan, 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."
Beginnings of a deal?
After initially demanding changes to or the elimination of Obama's signature health care reform plan, Republicans have more recently focused on extending the debt ceiling for up to six weeks while negotiating on spending and other issues during a continued government shutdown.
Democrats have insisted that the debt ceiling be raised and the government reopened before they would be willing to negotiate on other issues.
Some congressional Democrats have balked at the outline of the GOP offer from Thursday, insisting the government must reopen and the debt ceiling must be increased to get broader talks going.
"One way or another both of those have to happen," said veteran Sen. Carl Levin of Michigan.
Another Democrat, Rep. Elijah Cummings of Maryland, said Obama "needs to press for the opening of the government."
"Without a doubt the default would be much more catastrophic, but I've got constituents, a lot of whom work for the federal government who are going through catastrophes every hour," Cummings said.
Anti-Obamacare provisions no longer in GOP plans
One thing any agreement won't include, it appears, is a provision to defund the 2010 Affordable Care Act.
Ryan, who was the Republican party's vice presidential nominee last year, didn't mention Obamacare in a Wall Street Journal op-ed this week, saying instead that politicians from both parties should focus on "modest reforms to entitlement programs and the tax code."
As others have done in recent days, GOP Rep. James Lankford of Oklahoma told CNN on Thursday that cutting the funding for the President's signature health reform is "currently off the table."
However, Lankford said Republicans still seek a one-year delay in the penalties under Obamacare for people who fail to obtain health insurance, as required by the law.
According to a GOP source, it's not certain whether Boehner can gain support from some or most of his GOP caucus for a plan without anything to do with Obamacare or other concessions. That could mean, if a proposal like the one floated Thursday proceeds, the speaker may need Democratic votes to pass it.
Failure to raise the debt ceiling by next week's deadline would leave the government unable to borrow money to pay its bills for the first time in its history. And absent a breakthrough, the shutdown would continue at a cost estimated at up to $50 billion a month.
All of this is taking a toll on Washington's reputation: A national CNN/ORC International survey released Monday indicated that Americans are blaming all parties in the fight, though Republicans got the worst of it.