"There's a crack there," Republican Boehner said of the impasse in a brief interview near the end of a day of manoeuvring at the White House and the Capitol. But he added that it was not enough to warrant optimism.
Stocks fell significantly — the Dow Jones average by 159 points — as political gridlock endured. And, in the latest in a string of dire warnings, the International Monetary Fund said failure to raise America's debt limit could lead to default and disrupt worldwide financial markets, raise interest rates and push the U.S economy back into recession.
Republicans "don't get to demand ransom in exchange for doing their jobs," Obama said at the White House. "They don't also get to say, you know, unless you give me what the voters rejected in the last election, I'm going to cause a recession."
Even the deaths of U.S. servicemen over the weekend in Afghanistan were grist for the politicians. The Pentagon said that because of the partial shutdown it was unable to pay the customary death benefits to the survivors.
Boehner said Congress had passed and Obama signed legislation last week permitting the payments, adding it was "disgraceful" for the administration to interpret the measure otherwise. He said the House would clarify the issue with a new bill on Wednesday.
In Congress, a plan by Senate Democrats to raise the debt limit by $1 trillion to stave off a possible default drew little evidence of support from Republicans.
And a proposal by the House Republicans to create a working group of 20 lawmakers to tackle deficit issues drew a veto threat from the White House, the latest in a string of them as the administration insists the GOP reopen the government and avert default before any negotiations on deficit reduction or the three-year-old health care law can take place.
On a day in which both Obama and Boehner appeared on live television, both men appeared to be giving ground yet yielding little if anything of substance.
At midmorning, Boehner and other Republicans seemed to soften their demands.
"I suspect we can work out a mechanism to raise the debt ceiling while a negotiation is underway," said Rep. Tom Cole, an Oklahoma Republican who is close to Boehner.
The speaker, who had previously insisted on specific changes in the health care law as the price for preventing the shutdown, told reporters, "I want to have a conversation (with Obama and Democrats.) I'm not drawing any lines in the sand. It's time for us to just sit down and resolve our differences."
Asked if he was willing to raise the debt ceiling and fund the government for a short period, the Ohio Republican sidestepped. "I'm not going to get into a whole lot of speculation," he said.
A few hours later, Obama told a news conference he was willing to negotiate with Republicans on budget and other issues if Congress passed even short-term legislation to end the crisis.
"I'll even spring for dinner again," he said, referring to his courtship of Republican senators last winter, and attempting to inject humour into a political impasse where invective has been the norm.
Ninety minutes later, Boehner was unsmiling.
"What the president said today was if there's unconditional surrender by Republicans, he'll sit down and talk," he said. Renewing his call for "a conversation" about key issues facing the country, the Ohio Republican said, "Not next week. Not next month. The conversation ought to start today."
Boehner added, "The long and short of it is, there is going to be a negotiation here. We can't raise the debt ceiling without doing something about what's driving us to borrow more money and to live beyond our means."
Privately, officials said deeply suspicious Republicans were attempting to gauge Obama's comments to see whether they might represent a concession.
Treasury Secretary Jacob Lew has said the deadline for Congress to act is Oct., 17, setting that as the day the government will exhaust its ability to borrow funds and will have to rely day-to-day on tax and other receipts to pay its bills.
Some Republicans have downplayed the significance of the Oct. 17 deadline, saying that even then, the United States would be able to pay China and other holders of U.S. debt and avoid widespread economic dislocation.
But Obama said they were badly misguided, warning that default would harm the economy, cause retirement accounts to shrivel and houses to lose value. Still other Republicans have made it clear in recent days they agree with the threat posed by default and are determined to prevent it.
Inside a closed-door meeting of the Republican rank and file, Boehner had told his fellow Republicans they were in the midst of a tough battle and that Obama and Senate Majority Leader Harry Reid were trying to "annihilate us," according to one official in attendance.
Boehner's tone was different when he spoke to reporters.
"There's no boundaries here. There's nothing on the table. There's nothing off the table. I'm trying to do everything I can to bring people together and to have a conversation," he said.
In the back-and-forth, the threat of a default overshadowed the continuing partial government shutdown. An estimated 450,000 federal workers are idled at agencies responsible for items as diverse as food inspection and national parks, although all employees are eventually expected to receive full back pay.
The House approved legislation during the day to pay for a resumption of Head Start, the pre-school program for disadvantaged children. The vote was 248-168. The bill was the latest in a string of measures to end the shutdown in one corner of government or another in hopes of forcing Democrats to abandon their own demands for a full reopening of the federal establishment.
Republicans also announced they would vote to make sure federal workers on the job don't miss their next regularly scheduled paycheque on Oct. 15.
The shutdown began more than a week ago after Obama and Senate Democrats rejected Republican demands to defund "Obamacare," then to delay it, and finally to force a one-year delay in the requirement for individuals to purchase health care coverage or face a financial penalty.
It was not a course Boehner and the leadership had recommended — preferring a less confrontational approach and hoping to defer a showdown for the debt limit. Their hand was forced by a strategy advanced by Texas Sen. Ted Cruz and tea party-aligned House members determined to eradicate the health care law before it fully took root.
That portion of the strategy was doomed to failure, since money for the health care program was never cut off.
With the government partially shut down, Boehner and the GOP leadership decided to allow the closure showdown to merge with one over the debt limit.
Associated Press writers Donna Cassata, Andrew Taylor, Alan Fram, Martin Crutsinger, Julie Pace and Chuck Babington contributed to this story.