Washington (CNN) -- Angry Democrats rejected a proposal from Republicans Saturday for a two-step approach to raising the debt limit as congressional leaders raced against to clock to find a solution to the nation's debt-ceiling crisis, Democratic sources said.
Congressional leaders met again Saturday evening to discuss the two-step approach put forward by House Speaker John Boehner, which would raise the country's debt ceiling in stages, the sources said.
Leaders from both parties are anxious to reach a deal Sunday to head off a negative reaction in Asian markets when they open for the week, according to both Democratic and Republican aides.
Participating in the meeting were Boehner, Senate Majority Leader Harry Reid; Sen. Mitch McConnell, the Republican Senate leader; and House Democratic Leader Nancy Pelosi.
Under the two-step plan, the nation's debt ceiling would be raised through 2011 in exchange for spending cuts totaling around $1 trillion, the Democratic sources said. 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, they added.
While Democrats and Republicans agreed on the first stage of the plan, they split over how a second ceiling increase would be approved, the 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, they said.
"The Democrats who run Washington have refused to offer a plan," Boehner aide Michael Steel said soon after the meeting.
"Now, as a result, a two-step process is inevitable. Like the president and the entire bipartisan, bicameral congressional leadership, we continue to believe that defaulting on the full faith and credit of the United States is not an option," he said.
Pelosi similarly found fault with her counterparts, in a Saturday-night statement issued after the latest meeting.
"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 was even more targeted in his criticisms.
"I hope that Speaker Boehner and Leader McConnell will 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. We have run out of time for politics. Now is the time for cooperation," the Senate majority leader said in a statement.
Earlier Saturday, Boehner and other congressional leaders met with President Barack Obama in the White House, one day after Boehner broke off talks with the Obama administration on how to prevent a national default.
Boehner favors a plan to cut spending by $3 trillion to $4 trillion in the two-step process, a Republican aide said.
"He (Boehner) said he wanted to get something agreed to prior to Asian market trading (in the new week)," the aide told CNN. Asian markets are the first major market group to open worldwide each day.
Obama held a special one-hour meeting Saturday morning with congressional leaders following the collapse of direct talks between him and Boehner.
Participating in that meeting were Boehner, Reid, McConnell, Pelosi and Vice President Joe Biden. Treasury Secretary Tim Geithner and Jacob Lew, director of the Office of Management and Budget, joined after media cameras left the room.
The negotiations -- necessary to stave off an unprecedented national default that could prove economically devastating -- 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.
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.
Asked if Republicans pushed back on the idea of an extension through the end of 2012, the source said: "I just think it was an honest discussion about what might happen to this country if we default. It was like, everyone knows where everyone is at this point."
But 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."
Boehner aide Steel had earlier told reporters that "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. 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."
Reid, the Democrat Senate majority leader, reiterated Saturday afternoon that he sided with the president on rejecting any short-term extensions on debt-ceiling discussions.
"I will not support any short-term agreement, and neither will President Obama nor Leader Pelosi," Reid said in a statement. "We seek an extension of the debt ceiling through at least the end of 2012. We will not send a message of uncertainty to the world."
On Friday, Boehner walked away from debt talks with the Obama administration, saying he would take up the matter with congressional leaders.
"In the end, we couldn't connect. Not because of different personalities, but because of different visions for our country," Boehner wrote in a letter to his fellow Republicans.
Speaking to reporters after news of Boehner's decision broke, a visibly frustrated Obama said that he asked Republican and Democratic leaders in Congress to come to the White House on Saturday morning to "explain to me how we are going to avoid default" on the nation's debt.
The president said his administration had offered "an extraordinarily fair deal" to cut expenditures and raise revenues, in return for Congress agreeing to hike the nation's debt ceiling. But he said that Boehner "left (him) at the altar" by ending negotiations around 5:30 p.m. Friday.
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.
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, Jessica Yellin, Dan Lothian, Deirdre Walsh, Ted Barrett, Kate Bolduan, Alan Silverleib and Michael Martinez contributed to this report.