WASHINGTON – President Obama and House Republicans failed to reach agreement on a six-week extension of the nation’s borrowing authority during a meeting Thursday at the White House, but both sides agreed to keep talking and the Republican offer was seen as a first step toward ending the budget standoff.

Twenty Republicans, led by House Speaker John A. Boehner, R-Ohio, went to the White House at Obama’s invitation after a day of fine-tuning their offer to increase the Treasury Department’s authority to borrow money to pay existing obligations through Nov. 22.

In exchange, they sought the president’s commitment to negotiate a deal for long-term deficit reduction and a tax overhaul.

“The president didn’t say yes, didn’t say no,” said Rep. Paul D. Ryan, R-Wis. “We’re continuing to negotiate this evening.”

He added: “We put an offer on the table. We had a long, frank conversation about it. We agreed to continue talking and continue negotiating.”

Rep. Hal Rogers, R-Ky., chairman of the House Appropriations Committee, said Obama was hoping to reopen the government before going forward to negotiations.

“I think it’s clear that he would like to have the shutdown stopped, and that would require a CR,” Rogers said, referring to a continuing resolution that would finance the government. “And we’re trying to find out what it is that he would insist upon in a CR, and what we would insist upon. So our staffs are meeting, and we’ll talk later tonight.”

An initial report that Obama had rejected the Republicans’ offer was too definitive and came before Republican leaders or the White House had made it clear to reporters that the negotiations would continue.

The Republican proposal could come to a vote as soon as today. But the White House and congressional Democrats remained skeptical that House Republican leaders could pass the proposal.

A large faction of tea party conservatives campaigned on promises never to vote to increase the nation’s debt limit, and say they do not believe the warnings – including from Republican business allies – that failing to act could provoke a default and economic chaos globally. And House Democrats vowed not to support the proposal without a companion measure to fully fund a government now shuttered for 10 days.

Despite the president’s rejection, the House Republican offer represented a significant breakthrough. Ryan, chairman of the House Budget Committee, said Republicans were now willing to go to formal negotiations with Senate Democrats over a long-term, comprehensive budget framework, a move Republicans have resisted since April. And while House Republicans are divided over even a short-term increase in the debt ceiling, Rep. Tim Griffin, R-Ark., said the proposal would pass with GOP and Democratic votes.

Many Republicans said they would support their leadership’s proposal only if Obama agreed to not only negotiate a broader deal on a yearlong debt ceiling increase – which would most likely include spending and entitlement changes – but also negotiate over the terms for a stopgap spending measure that would reopen the government.

In a closed-door meeting Thursday morning, Republican lawmakers argued, alternately, for both a more moderate and a more hard-line approach. Rep. Charlie Dent, a Republican who represents a swing district in eastern Pennsylvania, stood up and pushed for a vote on a stopgap measure to finance the government with no strings attached, according to members who were in the room. But other more conservative members, Fleming said, are “concerned that we don’t move the goal posts unless we get the president to come to us, that this has to work bilaterally.”

Leaving the meeting early, Rep. Steve King, R-Iowa, one of his party’s more conservative members, said he was “not very enthusiastic” about the new strategy.

Still, despite continuing divisions within the Republican ranks and between the parties, the maneuvering suggested that congressional Republicans want to end a potentially calamitous standoff.

“When you’re genuinely convinced your own constituency – Democrats or Republicans – are beginning to waver, then that’s when you start closing down and moving forward,” said Rep. Jack Kingston, R-Ga. He said that had not happened yet, “but I do think that what reopens shutdowns more than policy are polls.”

Economists across a broad spectrum agree that violating the debt limit – which could happen as early as next week, absent congressional action – would probably severely damage the economy. The new Republican proposal could temporarily remove that threat.

Treasury Secretary Jacob J. Lew implored Congress on Thursday to raise the debt ceiling, warning the Senate Finance Committee of potentially severe market and economic repercussions if it did not.

In testimony before the committee, Lew stressed that the Treasury Department would run out of “extraordinary measures” to free up cash in a matter of days. At that point, the country’s bills might overwhelm its cash on hand in addition to any receipts from taxes or other sources, leading to an unprecedented default.

Lew said the Treasury had no work-arounds to avoid breaching the debt ceiling. “There is no plan other than raising the debt limit,” he said.

In not addressing the shutdown, Republicans would keep pressure on Democrats to negotiate on a package of deficit reduction and tax overhaul proposals. Griffin said the plan the speaker presented appealed to him because it would give House Republicans breathing room as they plot their next steps.

“We can’t even contemplate putting our full faith and credit, our credit rating and our financial system at risk, and this makes that clear,” he said.

House Republican leadership aides said the Nov. 22 date was picked to coincide with the Thanksgiving recess to add pressure on both parties to compromise. But with so little time, Boehner conceded, “you could end up back in the same place.”