Two days after breaking off talks with President Barack Obama on a $3 trillion-plus deficit reduction deal, House Speaker John Boehner said Sunday his last offer remains on the table as pressure mounted to avert a looming government default.

No formal negotiations were scheduled, but sources said that talks continued among staff members of both parties and the White House amid concerns that the continuing deadlock could prompt a downturn when Asian markets open trading for the week on Sunday night Washington time.

Boehner told “Fox News Sunday” he intended to propose a way forward on Sunday afternoon. He also planned a conference call with the House Republican caucus at 4:30 p.m., according to congressional aides.

Both Boehner, R-Ohio, and Treasury Secretary Tim Geithner said a comprehensive deficit deal that would bring Republican support for raising the federal debt ceiling was still an option.

“It may be pretty hard to put Humpty Dumpty back together again, but my last offer is still out there. I have never taken my last offer off the table,” Boehner said on “Fox News Sunday.”

Geithner told CNN’s “State of the Union” that the so-called “grand bargain” sought by Obama remains in play, but he also opened the door to a two-step process pushed by Republicans that would raise the debt ceiling with some spending cuts now, then bring broader structural and tax reforms next year.

“There’s nothing wrong with doing this in stages,” Geithner said, but he reiterated that a deal must remove the “threat of default” from the country by settling the debt ceiling issue through 2012 because “you don’t want politics messing around with America’s faith and credit.”

Geithner acknowledged the decision on increasing how much money the government can borrow in order to avoid defaulting on its obligations already has been politicized, saying, “They have taken this a little too far, frankly.”

If Congress fails to raise the $14.3 trillion debt limit by August 2, Americans could face rising interest rates, a declining dollar and increasingly jittery financial markets, among other problems.

Hard bargaining was likely to continue into the coming week, White House Chief of Staff Bill Daley warned on the CBS program “Face the Nation.”

“We may have a few stressful days coming up — stressful for markets of the world and the American people,” Daley said. “In the end, there’s no question in my mind, the American government will not default.”

Two senators involved in months of bipartisan talks on a separate deficit reduction plan by a group known as the “Gang of Six” expressed concern Sunday over the possibility of a government default.

“I’m not confident” of a deal to raise the debt ceiling, said Sen. Saxby Chambliss, R-Georgia, while Democratic Sen. Mark Warner of Virginia said failure to do so “would be the biggest act of irresponsibility in modern political history.”

At the Capitol building, a casually dressed David Krone, the chief of staff for Senate Majority Leader Harry Reid, was seen walking to Boehner’s office holding a slip of paper. About 20 minutes later, Krone emerged from the speaker’s office still holding his paper and munching on pizza.

Boehner announced Friday he was withdrawing from negotiations with Obama and instead would seek a deal with congressional Democrats, but Geithner said the president was involved in talks throughout Saturday that included congressional leaders from both parties.

On the Fox program, Boehner said the talks broke down last week because Obama and Democrats resisted “real cuts in spending” and demanded tax increases he opposes.

“After over six months of conversations with the president about doing the big deal, about taking a big step in the right direction, it is pretty clear to me that they are just not willing to do it — that the next election matters more than what’s right for the country,” Boehner said.

While insisting he would prefer to reach an agreement with Democrats, Boehner added: “If that’s not possible, I and my Republican colleagues in the House are prepared to move on our own, today.”

He offered no details, saying they were still being crafted. Any GOP-only plan would likely have trouble winning approval in the Democratic-controlled Senate.

According to congressional sources from both parties, a possible deal under discussion would involve two stages.

Under the plan, the nation’s debt ceiling would be raised through 2011 in exchange for spending cuts totaling around $1 trillion, according to Democratic sources. The ceiling would be raised again, through 2012, after a special commission is set up to find ways to reduce the long-term debt through entitlement and spending cuts and tax reform, the sources said.

While Democrats and Republicans agreed on the first stage of the plan, they split over how a second ceiling increase would be approved, the Democratic sources said.

Specifically, Democrats do not like the idea of tying future debt increases to a commission, which could deadlock and thrust the nation back into the uncertain position it is in today, according to the sources.

Both Geithner and Daley stressed the administration’s opposition to requiring two votes by Congress to increase the debt ceiling — one now and another in the first half of 2012, when the election season will be in full bloom.

When asked if Obama will follow through on his threat to veto such a deal, Geithner told CNN that Democrats in Congress would prevent such a measure from even reaching the president’s desk.

“It’s not going to make it that far… so that’s not a viable option,” Geithner said.

According to Boehner aide Michael Steel, a two-step process is “inevitable” because Democrats have failed to offer a concrete proposal.

The negotiations are testing the ability of leaders on both sides of the aisle to legislate effectively in an era of increasingly shrill and unyielding partisanship.

Republicans, who have railed against the growth of government, remain staunchly opposed to any tax increases. Democrats are trying to protect some of their party’s primary legacies — entitlements such as Social Security and Medicare, programs forged at the height of the New Deal and Great Society.

In a statement issued after a meeting among top congressional Democrats and Republicans on Saturday night, House Minority Leader Nancy Pelosi, D-California, criticized Republicans for refusing to accept Obama’s proposal to end Bush-era tax cuts on families earning more than $250,000 a year, which Democrats say represent the wealthiest 2 percent of the country.

“The delay in bringing forth a solution springs from the Republicans’ decision to walk away from 98 percent of the American people to protect the assets of the top 2 percent of the wealthiest people in our country,” she said in a statement.

Reid, D-Nevada, was even more targeted in his criticism of what he called Republican “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 in a statement. “We have run out of time for politics. Now is the time for cooperation.”

Earlier Saturday, Boehner and other congressional leaders met with Obama in the White House a day after Boehner broke off talks with the administration.

One Democratic official involved in the talks said the meeting was not contentious, and the participants did not rehash what went wrong with the Obama-Boehner talks. Rather it was very focused on “just how do we fix it,” and “everybody is pretty serious” about finding a way forward that prevents a default, the official said.

However, there was a recognition that the congressional leaders — all of whom say they want to prevent default — can only do so much about the opinions and actions of their caucuses, the official said.

“Different people put different ideas on the table” resulting in a mishmash of things that need to be sorted through to see what might be viable, the official said. Congressional staff will be sorting through the different ideas, the official added.

The president repeated his insistence that the debt ceiling be raised through the end of 2012, the source said. Both Geithner and Daley reaffirmed that stance on Sunday.

A House GOP aide told CNN that Republicans are “considering calling the president’s bluff” on his refusal to sign a bill that doesn’t raise the debt ceiling beyond the November 2012 election.

The aide said that party members are “struggling to see how they reach an agreement with significant debt reduction without buying time to work out the details.”

“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,” Steel, the Boehner aide, said earlier. “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.”

A Republican source familiar with the negotiations said Boehner told Republican lawmakers Friday that to get the debt ceiling raised by August 2, the House must vote on legislation by next Wednesday — and that means it must be posted online Monday.

CNN’s Brianna Keilar, Tom Cohen, Jessica Yellin, Dan Lothian, Deirdre Walsh, Ted Barrett, Kate Bolduan, Alan Silverleib, Matt Smith, Laurie Ure, Ana Sebescen and Michael Martinez contributed to this report.