A day after debt-reduction talks collapsed, congressional leaders arrived at the White House on Saturday morning for an emergency session to try to work out a deal with President Obama just 10 days before the government is set to run out of money to pay its bills.

Efforts to raise the legal limit on government borrowing were thrown into chaos Friday after House Speaker John A. Boehner abandoned talks with the White House over a dispute about taxes.

“We have now run out of time,” a visibly angry Obama said during an impromptu White House news conference held after Boehner (R-Ohio) called to say he was walking out on the talks for the second time in two weeks — again citing differences over taxes. Now, Obama said, “one of the questions that the Republican Party is going to have to ask itself is: Can they say yes to anything?”

Late Friday, a senior GOP aide said House leaders had “no idea” how to craft a plan to raise the $14.3 trillion debt limit that could win approval from the Democratic-controlled Senate and the Republican-led House. Talks were underway between Boehner and Senate leaders, but House Republicans continued to object to a developing Senate strategy to authorize Obama to raise the debt limit through 2013 without explicit congressional approval.

Ahead of the Saturday talks, a House GOP aide signaled that the speaker’s most likely position would be to push for a shorter-term deal. Both sides have identified more than $1 trillion in cuts, and Boehner’s camp suggested that some of those reductions could be used to meet the Republican demand of lifting the debt ceiling by cutting more than the dollar value of that increase in borrowing authority.

The president has repeatedly objected to any short-term deal, calling it “kicking the can down the road,” because there is a likelihood that the two sides would reach the same gridlock next year once such an extension was set to expire, and he did so again on Friday.

But on Saturday morning, the GOP aide said Obama was just trying to avoid dealing with the issue next year, when he will face reelection.

“Now, we do not know what size or shape a final package will take, but 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 reelection campaign.

“We want the most significant deficit-reduction possible, but linking the full faith and credit of the United States to presidential campaign politics is not a defensible position,” the aide added, speaking on the condition of anonymity to discuss Boehner’s strategy.

Boehner was clear Friday about the reasons for his decision to leave negotiations. The White House, he told reporters at his own news conference, “insisted on raising taxes.”

The collapse of the talks provided a dramatic end to a whirlwind week that began last Friday when Boehner alerted administration officials that he was ready to try again “to do something big” to control the debt, as a GOP aide put it. In the ensuing days, the two sides forged common ground on a two-stage strategy for raising the debt limit and cutting more than $4 trillion out of the federal budget through 2021.

That plan called for unprecedented cuts in agency spending, including at the Pentagon, and significant changes to Medicare and Social Security, the biggest drivers of future borrowing — a major concession for Obama and other Democrats. It also included an overhaul of the tax code that could have lowered personal and corporate income tax rates while reducing an array of popular tax breaks, such as the deduction for home mortgage interest, generating about $1 trillion in extra revenue over the next decade.

The package held the promise of taming the nation’s runaway debt.

The talks broke down, however, as the two sides tried to reach agreement on the magnitude of the tax and entitlement changes and how to force Congress to meet the goals of the plan. Boehner wanted a short-term increase in the debt ceiling that would have forced Congress to cut entitlement programs and rewrite the tax code before the limit could be increased again in February.

The White House rejected that approach and instead suggested a long-term debt-ceiling increase coupled with a trigger that would automatically cut taxes and entitlement spending if Congress did not enact promised changes in the tax code and entitlement programs. Boehner countered that the trigger should include the automatic repeal of key elements of Obama’s health-care reform, such as the requirement that everyone buy health insurance after 2014.

As the two sides tussled over the hardest ideological questions, administration officials said they tried to satisfy Boehner’s demands. But by Friday, the speaker concluded that the philosophical gulf between Republicans and the White House could not be bridged.

Boehner said Obama is “simply not serious” about cuts to Social Security and Medicare, unless they are accompanied by an increase of $1.2 trillion in tax revenue over the next decade — $400 billion more than the level set in a handshake agreement GOP aides said Boehner made with Treasury Secretary Timothy F. Geithner last weekend.

“Dealing with the White House is like dealing with a bowl of jello,” the speaker said. “There was an agreement with the White House for $800 billion in revenue. It was the president who walked away from this agreement.”

White House officials said that there was no handshake agreement on taxes, and acknowledged that they upped their revenue request after the bipartisan Senate “Gang of Six” released a plan to raise $2 trillion in taxes over the next decade. But they said Obama offered Thursday to drop the extra $400 billion if Boehner would accept smaller cuts to entitlement programs.

In the end, Obama told reporters he had offered Boehner “an extra-fair deal” on “the biggest debt-reduction package that we’ve seen in a very long time.” Obama said it would have raised taxes significantly less than the Gang of Six plan, which was endorsed by the third-ranking Republican in the Senate, Lamar Alexander (Tenn.).

Obama said he also put $650 billion in reductions to entitlement programs on the table, along with sharp cuts to government agencies, causing consternation among Democratic lawmakers and forcing him to “take a lot of heat from my party.”

“It is hard to understand why Speaker Boehner would walk away from this kind of deal,” Obama said as he launched into a barely controlled condemnation of “political posturing” and politicians who “dodge their responsibilities.”

House GOP leaders have repeatedly pulled the plug on efforts to forge agreement on a plan to control the national debt, Obama said. Democratic leaders had been willing to consider provisions that would have caused them considerable political pain among the elderly, unions and the party’s liberal base, he added. He also emphasized his willingness to compromise, noting that “I’ve been left at the altar now a couple of times.”

“This is not a situation where somehow this was the usual food fight between Democrats and Republicans. A lot of Democrats stepped up in ways that were not advantageous politically,” Obama said. He noted that he had offered to make further cuts to Medicare, even after Republicans denounced Democrats in last fall’s midterm campaigns for cutting Medicare as part of their overhaul of the health-care system, saying, “We’ve shown ourselves willing to do the tough stuff on an issue that Republicans ran on.”

Until Friday, when Boehner did not respond to several phone calls from the president, Obama said, “my expectation was that Speaker Boehner was going to be willing to go to his caucus and ask them to do the tough thing but the right thing” on raising taxes.

As policymakers braced for another full weekend of urgent meetings, Obama placed just one condition on the path forward, repeating his opposition to a short-term debt-ceiling extension that would force the two sides to replay the same battle in just a few months — an approach some House Republicans are considering. Instead, he insisted on an increase sufficient to pay the bills through the 2012 election — about $2.5 trillion.

Boehner, meanwhile, refused to back away from his demand that any increase in the debt limit be matched with spending cuts dollar for dollar — creating a huge potential obstacle.

The Senate’s top officials, Majority Leader Harry M. Reid (D-Nev.) and Minority Leader Mitch McConnell (R-Ky.), were at work on a fallback plan that would authorize Obama to raise the debt limit by $2.5 trillion in three installments without congressional fingerprints. Instead, Congress would disapprove of the increase, and Obama would be forced to veto their disapproval.

That would place the entire political burden on Obama, a burden the president said he is willing to accept. But Senate leaders have identified only about $1.5 trillion in spending cuts to pair with the debt-limit increase — a sum that falls far short of meeting Boehner’s dollar-for-dollar condition.

McConnell and Reid issued statements late Friday expressing their disappointment that talks between Obama and Boehner had collapsed, although both men had long been skeptical of the speaker’s ability to build political support for an ambitious deficit deal made behind closed doors with a president despised by many members of Boehner’s party. As they prepared to head to the White House on Saturday morning, both McConnell and Reid pledged to press forward.

“As I’ve said before, it’s time now for the debate to move out of a room in the White House and onto the House and Senate floors, where we can debate the best approach to reducing the nation’s unsustainable debt,” McConnell said.

The collapse of the talks came after financial markets closed, but could usher in great nervousness next week. Without additional borrowing authority, the Treasury could be unable to send out millions of Social Security checks on Aug. 3 or make a big payment due to investors days later. Obama said he has been discussing contingency plans with Geithner, who met Friday with Federal Reserve Chairman Ben S. Bernanke. The credit rating firm Standard & Poor’s has warned that not reaching a large-scale deal to reduce the debt could lead to an unprecedented downgrade of the nation’s AAA rating, a move that could send interest rates soaring.

Lawmakers have few other options for raising the debt limit. On Friday morning, the Senate defeated, in a party-line vote, a plan trumpeted by House and Senate conservatives that would have placed strict limits on federal spending and linked a debt-limit increase to a constitutional amendment requiring a balanced budget.

Reid declared that the proposal, called Cut, Cap and Balance, was “dead.” He also said he was setting aside a separate fallback plan he had been crafting with McConnell to permit the talks between Boehner and the White House to proceed.

“The path to avert default now runs first to the House of Representatives,” he said, adding later, “The Senate will wait anxiously.”

By lunchtime, however, a sense of foreboding descended on the Capitol as word spread that Boehner had not returned the president’s calls. Reid and McConnell spent Friday afternoon shuttling between each other’s offices, working to revive the plan they had just set aside.

Neither would comment on their talks. But Obama warned that they had better come up with a solution before the collapse of negotiations wreaks havoc on financial markets.

“It’s very important that the leadership understands that Wall Street will be opening on Monday,” he told reporters, “and we better have some answers during the course of the next several days.”