WASHINGTON - Congressional leaders raced Saturday to develop a new strategy for raising the federal debt limit that House Speaker John Boehner told his troops would include an ambitious plan to reduce future borrowing by as much as $4 trillion.

Although his talks with President Obama over a "grand bargain" to restrain the national debt collapsed in acrimony Friday, Boehner told House Republicans that he is confident lawmakers will avert a historic U.S. default -- a possibility just days off. "Over this weekend," Boehner said, "Congress will forge a responsible path forward."

Boehner and other leaders started their day at the White House, where Treasury Secretary Timothy Geithner warned of possible trouble in the markets if policymakers don't announce a viable plan for raising the debt limit before Asian exchanges open Sunday evening, said people familiar with the meeting. Aides said Geithner's warning lent fresh urgency to the talks, which continued throughout the day.

By evening, the outlines of a two-stage strategy were emerging. First, lawmakers would vote on a package to cut agency spending by as much as $1 trillion over the next decade and raise the debt limit, currently set at $14.3 trillion, by the same amount. That would give Geithner enough borrowing authority to cover the nation's bills through this year.

Then Congress would go to work to produce as much as $3 trillion in additional savings through an overhaul of the tax code and major changes to Social Security and Medicare.

Policymakers were still haggling over the process. One option is creating a new bipartisan debt-reduction panel comprising 12 lawmakers representing both the House and Senate, an idea offered this month by Senate Majority Leader Harry Reid, D-Nev. Another option is directing existing committees in Congress to make tax and entitlement changes aimed at meeting specific savings targets.

The two parties remained at odds, however, over how to force Congress to produce the second round of savings. Boehner proposed making the next debt-limit vote contingent on a plan to reduce future borrowing. But Democrats were insisting on a plan that would postpone another debt-limit showdown until after the 2012 election.

Boehner, Reid, Senate Minority Leader Mitch McConnell and House Minority Leader Nancy Pelosi met briefly in Boehner's office Saturday evening to try to resolve the issue. They emerged without a fresh approach to the problem, but Democratic aides described the meeting as productive and said the two sides agreed to keep working to produce a plan Sunday.

Obama has repeatedly objected to a short-term debt-limit extension, arguing that it would invite the same high-stakes gridlock that now threatens the sputtering U.S. recovery, as well as global markets. The president made that case again Saturday morning when Boehner, Reid, McConnell and Pelosi gathered at the White house for an hour-long session.

Obama called the meeting after Boehner walked out Friday on debt-limit talks with the White House for the second time in two weeks -- again citing differences on taxes.

"A short-term extension could cause our country's credit rating to be downgraded, causing harm to our economy and causing every American to pay higher credit card rates and more for home and car loans," said White House press secretary Jay Carney.

In the meeting, Pelosi supported the president. Reid later issued a statement saying he, too, would oppose a short-term deal. "We seek an extension of the debt ceiling through at least the end of 2012," Reid said. "We will not send a message of uncertainty to the world."

In his conference call with Republicans, however, Boehner said a short-term extension is among the options on the table, participants said. And he appeared to rule out the most likely alternative: A process proposed by McConnell this month that would authorize Obama to raise the debt limit without explicit congressional approval. Boehner said the plan "will be something new," according to a participant.

A senior GOP aide, meanwhile, accused Obama of trying to shape the debate to suit his political needs. "It would be terribly unfortunate if the president was willing to veto a debt-limit increase simply because its timing would not be ideal for his re-election campaign," the aide said.

But there were doubts about Boehner's ability to rally support for a debt-limit increase of any size among House Republicans, many of whom continue to push their plan to sharply cut spending over the next decade and adopt a constitutional amendment requiring Congress to balance the budget. Freshman Rep. Blake Farenthold, R-Texas, said Republican leaders remain concerned that even a small increase in the debt limit would fail on the House floor. "I think their concern about bringing it to the floor is whether they can get 218 [votes] or not," he said. "Everybody wants to only go through this pain once."