In a last-minute stab at compromise, Republican congressional leaders and the White House made significant progress Saturday toward a deal to avert a government default threatened for early next week, according to officials familiar with the talks.

Under the plan, the nation's debt limit would rise in two steps by about $2.4 trillion and spending would be cut by a slightly larger amount, these officials said. The first stage -- about $1 trillion -- would take place immediately and the second later in the year.

Congress would be required to vote on a balanced budget amendment to the Constitution, but none of the debt limit increase would be contingent on its approval.

One official said the two sides had settled on general concepts, but added there were numerous details to be worked out, and no assurance of a final agreement.

Senate Majority Leader Harry Reid, holding out hope for a last-minute compromise to raise the debt ceiling, abruptly postponed a key test vote late Saturday night to give negotiators “as much room as possible to do their work.”

The Democratic leader, speaking on the Senate floor, said he wanted to let a new round of White House-hosted talks run its course. He said the negotiators still have a “distance to go” and, to give them more time, announced the Senate would return for a vote at 1 p.m. ET Sunday.

That leaves just two days until the government hits an Aug. 2 deadline, after which the Obama administration says the Treasury will no longer be able to pay all its bills.

Reid did not explicitly endorse the latest round of talks or say whether he was a part of them. Republican congressional leaders had said earlier in the day that they were nearing a deal, but Reid disputed their account on the Senate floor. Republican leaders "should know that merely saying you have an agreement in front of television cameras doesn't make it so,” Reid.

Reid was tentatively planning to move ahead on his own deficit-reduction bill with a 1 a.m. ET test vote early Sunday. But Republican lawmakers were lining up in opposition to it.

In announcing his decision to postpone the vote, Reid said of the latest effort to reach a compromise: “I hope it bears fruit.”

He also said he doesn’t want to see a “short-term agreement” – meaning a deal that would only extend the debt ceiling for a limited amount of time.

House Speaker John Boehner and Senate Republican Leader Mitch McConnell said Saturday that they are confident they can reach a deal with the White House to raise the nation's $14.3 trillion debt limit to allow the government to keep paying all of its bills.

At a news conference held just minutes after the GOP-led House defeated a version of Reid’s debt-limit bill, McConnell said he had spoken with President Obama and Vice President Biden in the past hour.

"I'm confident and optimistic that we're going to get an agreement in the very near future and resolve this crisis in the best interest of the American people," he said.

"Our country is not going to default for the first time in history," McConnell said."We now have a level of seriousness with the right people at the table. ....We're going to get a result."

Boehner added he's also confident of an agreement with the White House "to end this impasse."

But Reid disputed their account after meeting with the president and House Democratic leader Nancy Pelosi at the White House.

"Republican leaders still refuse to negotiate in good faith," Reid said, explaining that they still refuse to consider including new revenues in any deal and only want to slash entitlement programs.

McConnell responded that he's more optimistic than Reid and that the only way to get to an agreement before Tuesday is through the president.

"We need to be in a position where all of us in the leadership can come back here and say that we think we reached a framework of an agreement that we can recommend to our members," he said. "So that's what I'm working on and I'm not interested in scoring any political points. I'm interested in getting an outcome for the American people. And the only way that can be done is with the president of the United States."

Earlier Saturday, the House defeated Reid's bill that would raise the nation's $14.3 trillion debt limit by $2.4 trillion while cutting spending by $2.2 trillion. But the Senate hasn't voted on the bill yet.

In his weekly radio and Internet address Saturday, the president pushed congressional leaders to reach a deal.

"There is very little time,"Obama said. He called for an end to political gamesmanship, saying, "the time for compromise on behalf of the American people is now."

Boehner said that despite the partisanship of recent weeks, "I think we're dealing with reasonable, responsible people who want this crisis to end as quickly as possible and I'm confident it will."

The Associated Press contributed to this report.