"Let's put this hysterical talk of default behind us and instead start talking about finding solutions," said Senate Republican Leader Mitch McConnell of Kentucky.
Republicans in the House and Senate separately made proposals to the White House for ending an impasse that polls say has inflicted damage on their party politically.
Each offered to reopen the government and raise the $16.7 trillion debt limit — but only as part of broader approaches that envision deficit savings, changes to the health care law known as Obamacare and an easing of across-the-board spending cuts that the White House and Congress both dislike. The details and timing differed.
"We're waiting to hear" from administration officials, said House Majority Leader Eric Cantor.
Hopes remained high on Wall Street, where investors sent the Dow Jones industrial average 111 points higher following Thursday's 323-point surge.
In meetings with lawmakers over two days, Obama left open the possibility he would sign legislation repealing a medical device tax enacted as part of the health care law. Yet there was no indication he was willing to do so with a default looming and the government partially closed.
Obama called Boehner at midafternoon, and Michael Steel, a spokesman for the leader of House Republicans, said, "They agreed that we should all keep talking."
Jay Carney, the president's press secretary, said Obama "appreciates the constructive nature of the conversation and the proposal that House Republicans put forward. At yet, the spokesman said, "He has some concerns with it."
In Congress, the man certain to be involved in any final agreement, Senate Majority Leader Harry Reid, gave no indication of his plans.
While the impact of the shutdown varies widely, lawmakers seemed to be taking care of their own needs.
The members-only House gym remained in operation, and enough Senate staff was at work to operate the aging underground tram that ferries senators and others from the Russell Office Building to the Capitol a short distance away.
The shutdown sent ripples nationwide.
The aerospace industry reported that furloughs at the Federal Aviation Administration have resulted in a virtual stop to certification of new aircraft, equipment and training simulators.
The Senate passed legislation instructing the Pentagon to permit military chaplains to conduct worship services. House approval was still needed.
And Keith Colburn, a crab fisherman, told lawmakers during the day that a lucrative, one-month crab harvest set to begin Oct. 15 in the Bering Sea is in jeopardy because the National Oceanic and Atmospheric Administration is not assigning quotas to boats.
Obama met at the White House for more than an hour with Senate Republicans, the last in a series of four presidential sit-downs with the rank and file of each house and each party.
He has said repeatedly in recent weeks that he is willing to negotiate with Republicans on budget, health care or other issues, but only after the government is reopened and the threat of default eliminated.
The White House seemed to wobble on that point on Thursday, until Senate Majority Leader Reid emphatically reinforced that it was his view, too.
Republicans have just as insistently demanded that Obama negotiate with them in exchange for passage of legislation that both sides agree is essential.
That left the White House and congressional leaders looking for a way to negotiate their way out of an impasse without appearing to negotiate — with the health of the nation's economy dependent on their political dexterity. The administration says the government will bump up against its borrowing limit next Thursday, raising the spectre of an unprecedented default.
At Obama's meeting with Senate Republicans, Sen. Susan Collins of Maine laid out a proposal to raise the debt limit until the end of January, reopen the government and take a slice out of the health care law.
Under a proposal she and other GOP senators have been developing, a medical device tax that helps finance the health care law would be repealed, and millions of individuals eligible for subsidies to purchase health insurance under the program would be subject to stronger income verification.
At the same time, federal agencies that have been affected by across-the-board cuts would gain greater flexibility in the use of their remaining funds.
Any other items could be negotiated later.
Back at the Capitol, Collins said Obama said the proposal "was constructive, but I don't want to give the impression that he endorsed it."
For their part, House Republicans previewed a different approach in a late-night meeting Thursday with White House officials.
It, too, would raise the debt limit and avoid a default, as part of a framework that could include easing the across-the-board cuts in exchange for reductions that Obama has supported in the past in benefit programs. That plan, too, seeks changes in Obamacare.
White House officials declined to comment on that proposal, although administration aides were checking with key Democrats in Congress to gauge their reaction.
The White House and Republicans have negotiated almost $4 trillion in deficit savings in the past three years. But little of that has come out of benefit programs such as Medicare and Medicaid, and the Republican proposals seemed an attempt to open up that part of the budget to scrutiny.
Obama has proposed raising the cost of Medicare for better-off seniors, a $50 billion item over a decade. He has also backed higher fees under TRICARE, which provides health care for nearly 10 million active-duty and retired military personnel, retirees, reservists and their families, as well as increases in the cost of retirement benefits for federal workers.
After four years of trillion-dollar deficits, the 2013 federal budget shortfall is expected to register below $700 billion, but Republicans say more cuts are essential. At the same time, the nation's debt is rising inexorably — the reason for the effort to raise borrowing limit to cover it. The debt was $10.6 trillion when Obama took office during the worst recession in decades, and has grown by $6.1 trillion in the years since.
Associated Press writers Julie Pace, Jim Kuhnhenn, Andrew Taylor, Alan Fram and Donna Cassata contributed to this report.