House Speaker John A. Boehner and Treasury Secretary Timothy F. Geithner appeared on the political talk shows Sunday at the start of a day of crucial talks, and their comments confirmed that the gap between Republicans and the White House on the debt ceiling remains wide and deep.

Geithner warned that a debt-reduction deal that doesn’t result in raising the nation’s legal borrowing limit through the 2012 elections remains unacceptable to the White House and cannot win enough Democratic votes in Congress to ensure passage.

Boehner (R-Ohio), meanwhile, indicated that Republicans could push forward with a plan this week, even without Democratic support.

“I think the preferable path would be a bipartisan plan that involves all the leaders, but it’s too early to decide whether that’s possible,” he said on “Fox News Sunday.” “If a bipartisan bill is not possible, I and my Republican colleagues in the House are prepared to move on our own.”

Under proposals floated by Republicans on Saturday, Congress would cut agency spending by as much as $1 trillion over the next 10 years and raise the debt ceiling by an equal amount. That would give the government the borrowing authority to pay its bills, but only through the end of this year.

The deal would then require Congress to work to reduce the debt by as much as $3 trillion by reforming the tax code and entitlement spending.

Geithner said such a two-step process could be acceptable — but not if it requires a second vote by Congress to lift the debt ceiling in the coming months. He insisted that any deal must raise the limit long enough to remove the possibility of a default from the politically heated presidential campaign season.

“There’s nothing wrong with doing this in stages,” he said on CNN’s “State of the Union.” “But what it cannot do is leave the threat of default hanging over the economy.”

White House Chief of Staff William M. Daley had a similar message on NBC’s “Meet the Press,” saying a “short-term gimmick” won’t work and that President Obama would veto any plan that didn’t raise the ceiling through the 2012 elections.

On ABC’s “This Week,” Geithner said a deal must be in motion in the House by Monday night to avoid missing the Aug. 2 deadline.

“They need to have a framework that they know with complete confidence will pass both houses of Congress, that is acceptable to the president, and that should happen today,” he said.

Congressional leaders had raced Saturday to develop a new strategy for raising the debt limit that 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 said he was confident lawmakers would avert a historic U.S. default — a possibility just 10 days off.

“Over this weekend, Congress will forge a responsible path forward,” Boehner said in a statement.

The speaker and other leaders started their Saturday at the White House, where Geithner warned of possible trouble in the markets if policymakers don’t announce a viable plan for raising the debt limit before the Asian exchanges open Sunday evening, according to people familiar with the meeting. Aides said Geithner’s warning lent fresh urgency to the negotiations, which continued throughout the day on Capitol Hill.

By early 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 the end of 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, the biggest drivers of federal spending. To identify those savings, Congress would create a new bipartisan debt-reduction committee comprising 12 lawmakers from both the House and Senate, an idea offered by Senate Majority Leader Harry M. Reid (D-Nev.).

But the two sides were still fighting over how to force Congress to produce the second round of savings. Boehner wants to set another debt-limit vote early next year, while Democrats are insisting on a plan that would postpone another debt-limit showdown until after the 2012 presidential election.

Boehner, Reid, Senate Minority Leader Mitch McConnell (R-Ky.) and House Minority Leader Nancy Pelosi (D-Calif.) met briefly in Boehner’s office Saturday evening to try to break the impasse. They emerged without a fresh approach to the problem, and aides made plans to try again Sunday morning.

After the meeting, Reid issued a statement saying he was “deeply disappointed in the status of negotiations” and urging Republicans “to reconsider their intransigence.”

“Their unwillingness to compromise is pushing us to the brink of a default on the full faith and credit of the United States,” Reid said. “We have run out of time for politics. Now is the time for cooperation.”

Boehner spokesman Michael Steel fired back in a statement that Democrats have long been aware of “our principles to pass a bill that fulfills the President’s request to increase the debt limit beyond the next election. . . . The Democrats who run Washington have refused to offer a plan. Now, as a result, a two-step process is inevitable.”

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. The president made that case again Saturday 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 for the second time in two weeks — again citing differences over 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,” White House press secretary Jay Carney said in a statement. “As the current situation makes clear, it would be irresponsible to put our country and economy at risk again in just a few short months with another battle over raising the debt ceiling.”

All three major rating agencies have threatened to downgrade the United States if the debt limit is not raised. So far, however, the market for Treasury bonds has shown little distress over the possibility of a U.S. default, despite numerous snags in negotiations.

In a conference call with House Republicans, Boehner ruled out the most likely alternative to a short-term extension: a process proposed by McConnell this month that would authorize Obama to raise the debt limit in installments, without explicit congressional approval.

“The goal of our discussions now with the leaders is not to do something in the Reid-McConnell framework,” Boehner said, according to one participant. “It will be something new.”

A senior GOP aide, meanwhile, accused Obama of trying to shape the debt-limit debate to suit his political needs, saying, “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.”

Meanwhile, there were lingering doubts about Boehner’s ability to rally support for a debt-limit increase of any size or duration. Many House Republicans continue to push their plan to sharply cut spending over the next decade and adopt a constitutional amendment requiring Congress to balance the budget. Such a plan passed the House, but failed Friday in the Senate on a party-line vote.

Freshman Rep. Blake Farenthold (R-Tex.) 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,” Farenthold said in an interview. “Everybody wants to only go through this pain once.”

On the conference call, Boehner assured his colleagues that, despite a week of secret talks with Obama, “there was never a deal” to cut spending and raise taxes by $800 billion over the next decade, as Boehner himself suggested in a news conference Friday night. “Any new revenue would have come from tax reform and economic growth,” Boehner said, according to participants — “not a tax hike.”

The terms of the emerging deal changed, however, when Obama “demanded 50 percent more revenue,” Boehner said. “That looked like a tax hike. We said no.”

Rank-and-file Republicans praised Boehner for “standing firm” and “doing the right thing,” participants said. “The American people are on your side,” said Rep. Ann Marie Buerkle (R-N.Y.).

Administration officials acknowledge that they requested more revenue last Wednesday in response to Boehner’s request for more cuts to federal health programs. But they said the request for an extra $400 billion was never a demand or an ultimatum. It was a negotiating position, to which they expected Boehner to respond.

In a meeting Thursday morning, a Democrat familiar with the talks said Boehner’s top policy aides sat for nearly two hours with White House officials without indicating that the extra taxes were a deal breaker. Then, the Democrat said, the speaker’s office simply stopped calling.

While Boehner and Obama sparred over the extra $400 billion in taxes, some Republicans were focused on the $800 billion that Boehner was ready to give up. One GOP Senate aide called it “a huge concession.” Another said it would open the door to a rewrite of the tax code that would guarantee significant tax hikes for households at all income levels.

The senior Republican on the tax-writing Senate Finance Committee, Sen. Orrin Hatch (R-Utah), has been warning for months against such a move, which has also been championed by Obama’s fiscal commission and a bipartisan group of senators known as the Gang of Six.

“I am open to looking at eliminating or reducing some tax expenditures as part of comprehensive tax reform, but only if tax rates are lowered enough” to ensure that the government does not collect more tax revenue, Hatch said in a recent speech on the Senate floor. If tax reform raises any more money — much less an extra $800 billion — Hatch said, “that is a tax increase, plain and simple.”

Staff writers Paul Kane, David Fahrenthold, Rosalind S. Helderman, Felicia Sonmez and Zachary A. Goldfarb contributed to this report.

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