With the default deadline just two days away, prospects for a debt-limit deal seemed to significantly brighten early Sunday after the White House entered intense negotiations with Senate Minority Leader Mitch McConnell on Saturday in a last-ditch bid to forge a bipartisan agreement to raise the federal debt limit.

Late Sunday afternoon, Senate Majority Leader Harry M. Reid (D-Nev.) had given his approval to the emerging debt deal, but must still present the plan to his caucus possibly later Sunday evening. “Senator Reid has signed off on the debt-ceiling agreement pending caucus approval,” his office said.

After emerging from a lengthy closed-door meeting with House Minority Leader Nancy Pelosi (D-Calif.) and other Democratic leaders late Sunday afternoon, Reid (D-Nev.) was asked whether he expects a vote on the deal tonight.

“I hope so,” he told reporters.

But sources in both parties said all sides were close to signing off on the emerging agreement except House Speaker John A. Boehner (R-Ohio). While the reason for the hangup was not immediately clear, the framework has the potential to trigger big cuts in the defense budget. Defense cuts are likely to be problematic for many of the veteran Republicans Boehner will need to push a compromise through the House. A Boehner spokesman did not immediately respond to questions.

Meanwhile, as expected, the Senate failed to break the 60-vote threshold to end debate on a bill by Reid’s (D-Nev.) bill to raise the debt ceiling by $2.2 trillion in an early afternoon vote on Sunday as the outlines of the deal between Senate leaders and the White House began to take shape. Fifty senators voted to end debate on Reid’s bill and 49 opposed stopping it.

The winds began shifting towards a deal shortly after 10 p.m. Saturday night when Reid announced that talks between McConnell (R-Ky.) and Vice President Biden had made significant progress, prompting Reid to delay a vote that had been scheduled for 1 a.m. Sunday on his own debt-limit measure.

“Relief” is the way Sen. Chuck Schumer (D-N.Y.) described the mood on Capitol Hill on CNN’s State of the Nation Sunday morning. “Default is far less of a possibility now then it was even a day ago, because the leaders are talking and talking in a constructive way.”

But, Schumer added, “There are lots of things that are not filled in.”

On the same show, McConnell echoed that “we had a good day yesterday” and the parties “were very close” to crafting a deal framework that he could “soon” recommend that Senate Republicans support.

“We’re not going to have default,” McConnell stated.

“There are negotiations going on at the White House to avert a catastrophic default on the nation’s debt,” Reid said, after announcing that the vote had been pushed back to 1 p.m. Sunday. “There are many elements to be finalized, and there is still a distance to go before any arrangement can be completed. But I believe we should give everyone as much room as possible to do their work.”

He added: “I’m glad to see this move toward cooperation and compromise. I hope it bears fruit.”

The emerging agreement calls for raising the $14.3 trillion debt limit by up to $2.4 trillion in two stages, with the debt limit rising unless two-thirds of both chambers of Congress disapprove, according to officials in both parties familiar with the talks. That would extend the Treasury’s borrowing authority into 2013, satisfying President Obama’s demand to avoid another showdown over the issue in the heat of the 2012 presidential campaign.

The first stage would pair an increase of roughly $1 trillion with cuts to government agencies of about the same magnitude over the next decade. In the second stage, a special congressional committee would be created to identify additional savings later this year. The size of those savings would dictate the size of the second debt-limit increase, giving Republicans the dollar-for-dollar matchup they have demanded between spending cuts and the debt-limit increase.

The focus of talks Saturday was a mechanism to force the committee to act and to ensure that Congress adopts its recommendations. Late Saturday, McConnell and Biden appeared to have settled on a trigger that would cut most government spending – including the defense budget and Medicare — by as much as 4 percent if the committee failed to produce a plan to tame the spiraling national debt, the officials said.

It was not immediately clear whether Boehner would accept the emerging framework, or whether he could persuade his fractious conference to support it. But Boehner, too, showed signs of paving the way for compromise Saturday, convening a meeting of the Tuesday Group, a bloc of about 45 moderate Republicans whose votes would be key to pushing any deal through the House.

“I’d like a big, bold, bipartisan plan. I think that’s what the country needs,” Rep. Robert J. Dold (R-Ill.) said after the meeting. “It’s certainly something I’m looking for.”

Reid’s announcement of progress in the talks shattered an air of desperation that had settled over the Capitol earlier in the day as rank-and-file lawmakers pleaded with their leaders to set politics aside and strike a bipartisan deal. After months of negotiation and weeks of debate, every black-and-white plan for cutting government spending and raising the $14.3 trillion debt ceiling was doomed or had already been shot down.

As the battle entered its final hours, the nation’s leaders were left bickering among themselves, unable to agree even on a modest version of the deal they said they were close to sealing a week ago. Their remaining differences were not especially large, but a resolution appeared elusive.

“It really shouldn’t be this difficult,” Sen. John F. Kerry (D-Mass.) said in a speech on the Senate floor. “I believe there’s a bipartisan consensus here in the Senate . . . if we stop talking past each other.”

Throughout the afternoon and into the evening, Reid and McConnell conducted separate talks with the White House in search of an agreement. But the two men were barely speaking to each other, and their anger boiled over onto the Senate floor.

Reid accused McConnell of blocking the path to compromise that many Senate Republicans were eager to pursue.

“While the Republican leader is holding meaningless press conferences, his members are reaching out to me . . . to try to move the process forward,” Reid said testily, referring to a midafternoon news conference when McConnell and announced that they were in direct talks with Obama.

McConnell fired back that he was “more optimistic than my friend, the majority leader,” after negotiating privately with Biden. “I think we’ve got a chance of getting there,” McConnell said.

“We’re going to get a result on a bipartisan basis. That’s what I’m working on,” McConnell added, complaining that he had been forced to “cut short a conversation with the vice president” to answer Reid’s summons to a meaningless procedural vote. “I’d actually like to get back to work.”

A Democratic official familiar with the talks said Biden and McConnell were focused on combining the debt-limit plans that emerged from bipartisan negotiations among congressional leaders last weekend. That effort was cut short by Democrats, including Obama. The next day, Reid and Boehner unveiled very similar bills that would cut deeply into agency budgets and create a 12-member committee to find more savings by the end of the year.

The key difference was the length of the debt-limit extension. Boehner’s bill, which passed the House but died in the Senate, would have granted a reprieve only through February or March 2013, after the 2012 election is decided. Unless the committee came up with $1.8 trillion in additional savings, Congress would face another standoff over the debt limit.

Reid’s bill, which was pending in the Senate on Saturday, would increase the debt limit into 2013 and count $1 trillion in savings from winding down the wars in Iraq and Afghanistan, a move Republicans decried as an accounting gimmick.

To bridge the divide, McConnell and administration officials had been trying all week to design a mechanism to force the committee to act, giving Republicans their cuts and Obama his debt-limit increase.

Republicans were opposed to a trigger that would force automatic tax increases; Democrats were opposed to a trigger with spending cuts only.

On Saturday, Democrats appeared to have conceded the point in exchange for big automatic defense cuts, which they said would give Republicans a powerful incentive to work with Democrats toward a more palatable compromise.

Unless Congress lifts the debt limit by Tuesday, Treasury officials have said they will begin to run out of cash to pay the nation’s bills. That could force the government into default, an outcome that could devastate the sputtering U.S. recovery and global financial markets. Over the past week, the Dow Jones industrial average fell nearly 540 points amid rising anxiety about the debate in Washington.

As talks with the White House got underway Saturday, McConnell appeared briefly with Boehner to assure the nation that a resolution was at hand.

“Our country is not going to default for the first time in history; that is not going to happen,” McConnell said. “We now have a level of seriousness, with the right people at the table.”

All day Saturday, however, both parties were on the attack. In the House, Republicans were smarting over the Senate’s quick Friday night defeat of Boehner’s proposal for a short-term debt-limit increase. So the GOP returned the favor with a unanimous vote in opposition to Reid’s proposal to increase the debt limit into 2013.

It was not a real vote. The Reid measure was still pending before the Senate. So Rep. David Dreier (R-Calif.), chairman of the House Rules Committee, drew up his own version of the Reid bill and sent it to its death on the House floor, where 11 Democrats joined Republicans in rejecting the measure, 246 to 173.

“There are some good things in Senator Reid’s proposals. But I will say this: I don’t believe that continuing down the road towards increasing debt ceilings without the kinds of checks necessary is the right thing to do,” Dreier said.

Even by House standards, the debate was rowdy and sharp-edged, as lawmakers in both parties hooted and shouted down their colleagues across the aisle. Republicans booed Minority Leader Nancy Pelosi (D-Calif.), who accused Boehner of going “to the dark side” when he rewrote his debt-limit bill to add required passage of a balanced budget amendment — a change designed to appeal to his far-right wing.

Pelosi turned toward her GOP colleagues and said it again with emphasis. “Let me repeat,” she said, “he chose to go to the dark side.”

The atmosphere was just as tense and bitter in the Senate, where McConnell delivered a letter to Reid signed by 43 Republicans declaring their opposition to his version of the debt-limit bill.

Four Republican senators did not sign the letter: Olympia Snowe and Susan Collins of Maine, Scott Brown of Massachusetts and Lisa Murkowski (R-Alaska). But there was little evidence that Reid’s bill could muster the 60 votes needed to overcome a Republican filibuster — a fact Reid tacitly acknowledged when he rejected McConnell’s offer to stage a vote early Saturday evening.

“There are some good things in Senator Reid’s proposals. But I will say this: I don’t believe that continuing down the road towards increasing debt ceilings without the kinds of checks necessary is the right thing to do,” Dreier said.

In his weekly address, Obama repeated his call for a bipartisan compromise.

“It must have the support of both parties that were sent here to represent the American people — not just one faction of one party,” the president said. “Congress must find common ground on a plan that can get support from both parties in the House. And it’s got to be a plan that I can sign by Tuesday.”

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Rachel Van Dongen contributed to this report.