“We’ll be working that out over the next couple of months,” Mr. Boehner said in a comment that understated just how difficult that job was going to be.
Six months after becoming speaker and six weeks before the government runs out of borrowing power, Mr. Boehner will now be one of the people “working that out” directly with President Obama and his fellow Congressional leaders. Bipartisan budget talks imploded on Thursday when Representative Eric Cantor, the majority leader, took a walk.
The speaker faces quite a predicament. He has to navigate his own party’s internal politics, cut a deal with his cagey counterparts in the Senate and Mr. Obama, and steer any agreement through the House where many Republicans are sure to oppose it — all while trying not to sink the entire economy in the process.
The challenges might explain why Mr. Cantor, after a contentious negotiating session on Wednesday, evidently decided the job was above his pay grade and passed the debt-limit baton to Mr. Boehner.
Trying to get the negotiations back on track on Friday, Mr. Obama invited Senators Harry Reid, the Nevada Democrat and majority leader, and Mitch McConnell of Kentucky, the Republican leader, to the White House on Monday for separate sessions to talk about the debt limit.
Jay Carney, the White House press secretary, said the meetings would be an effort to “find common ground on a balanced approach to deficit reduction.”
The crucial phrase there is the reference to a balanced approach. On Friday, Democrats emphasized anew that they could not accept any compromise tied to a debt-limit increase that relied solely on spending cuts and that did not provide some new federal revenue from taxes on businesses and the most affluent Americans.
“The simple message that Republicans sent yesterday by walking out of the talks was that unless we take their lopsided approach to reducing the deficit, they are prepared to tank the economy,” said Representative Chris Van Hollen of Maryland, the top Democrat on the Budget Committee and a participant in the negotiations. Mr. Van Hollen said Mr. Cantor and Senator Jon Kyl of Arizona, the other Republican who had been in on the negotiations, refused to consider such revenue sources as eliminating a tax benefit for owners of corporate jets.
Mr. Cantor cited persistent Democratic insistence on raising tax revenues as the main reason for his decision to abandon the talks. Mr. Boehner and Mr. McConnell have been equally crystal clear that they are opposed to including anything that even resembles a tax increase in a budget compromise. So it remains to be seen how Mr. Obama clears that hurdle after Vice President Joseph R. Biden Jr., who was leading the first round of talks, was unable to do.
Other factors contributed to the abrupt end of the Biden negotiations as well. Democrats and some Republicans viewed Mr. Cantor’s departure as a reaction to his learning that Mr. Boehner, with whom he has a sometimes competitive relationship, was at the White House on Wednesday for a meeting with Mr. Obama that had not been publicly announced. Their thinking was that Mr. Cantor was miffed that the speaker, without informing him, was already engaged in sidebar negotiations with the White House.
A Cantor aide dismissed such reasoning as silly and said the talks were always going to move to the presidential-speaker-Senate leader level.
“The bottom line is that the proposals pushed by Democrats would increase taxes by hundreds of billions of dollars on individuals, small businesses, and employers at a time when we need to focus on job growth,” said Brad Dayspring, a spokesman for Mr. Cantor.
Officials said that in what turned out to be a final negotiating session on Wednesday, Mr. Cantor was pushing for more reductions in annual federal spending while refusing to consider cuts in the Pentagon budget — a sore point with Democrats. The impasse over the spending total — and the gulf between Democrats and Republicans on any new tax revenues — scuttled the talks for now.
But participants and aides said the Biden sessions did produce a lot of budget material that will be useful in the meetings between Congressional leaders and the White House.
White House officials started laying out their argument for new revenues on Friday by trying to define the choices to the administration’s advantage. They described the tax increases as closing loopholes for “millionaires and billionaires,” while saying the deep spending cuts that the Republicans’ approach would force would hamper essential government services while driving up health costs for Medicare recipients.
Speaking at a fund-raiser in New York on Thursday night, Mr. Obama offered his view of the standoff, saying: “This is not just a budget question, this is a values question.”
White House officials say the ideal balance in a package to reduce the deficit by about $2 trillion over a decade would be $1 of new tax revenues for every $3 in spending reductions, including lower interest on the debt that would be automatic if projected deficits are reduced.
The next phase of the talks promises to be difficult. But the spending agreement negotiated earlier this year showed that the president, Mr. Boehner and Mr. Reid can seal a deal. Still, Mr. Boehner made clear on Friday that he would not be part of an agreement that raised taxes or did not enact budget overhauls and cut spending by at least as much as the debt ceiling was raised.
“If the president and his allies want the debt limit increased,” he said, “it is only going to happen via a measure that meets these tests.”
Jackie Calmes contributed reporting.