Vice President Biden and Sen. Mitch McConnell were locked in urgent talks late Sunday over the “fiscal cliff” after Democrats offered several significant concessions on taxes, including a proposal to raise rates only on earnings over $450,000 a year.

With a New Year’s Eve deadline hours away, Democrats abandoned their earlier demand to raise tax rates on household income over $250,000 a year, as President Obama vowed during the recent presidential campaign.

They also relented on the politically sensitive issue of the estate tax, according to a detailed account of the Democratic offer obtained by The Washington Post, promising to stage a vote in the Senate that would guarantee that taxes on inherited estates remain at their current low levels, a key GOP demand.

Still, McConnell (R-Ky.), the Senate minority leader, was holding out to set the income threshold for tax increases even higher, at $550,000, according to people close to the talks in both parties. And he was protesting a Democratic proposal to raise taxes on investment profits for households with income above $250,000.

The two sides were also sharply at odds over automatic spending cuts set to decimate budgets at the Pentagon and other federal agencies next month. Democrats were seeking to delay the cuts, known as the “sequester,” until 2015, without identifying other savings to compensate. They were also pressing to extend unemployment benefits, farm subsidies and Medicare payments to doctors, again without offsetting cuts as Republicans demand.

Unless the two sides can reach agreement, historic tax hikes are set to hit virtually every American on Jan. 1, potentially driving the nation back into recession. An impasse would also throw the coming tax filing season into chaos, as nearly 30 million unsuspecting taxpayers would be required to pay the costly alternative minimum tax for the first time.

As Biden and McConnell traded phone calls deep into the night, lawmakers waited anxiously for news. Though members of both parties received lengthy briefings from their respective leaders about the status of the talks, senators were just as likely to predict that the nation was on the verge of a self-inflicted economic crisis as they were to predict that salvation was at hand.

“I think we’re going over the cliff,” Sen. Lindsey O. Graham (R-S.C.) wrote on Twitter in the middle of the day.

“The two parties are so close that they can’t afford to walk away,” Sen. Mike Johanns (R-Neb.) countered hours later. “I continue to be optimistic.”

Biden, a veteran dealmaker who served in the Senate for 36 years, entered the talks Sunday at McConnell’s request after the Republican leader said he had grown “frustrated” by the pace of negotiations with Senate Majority Leader Harry M. Reid (D-Nev.).

Personal relations between the two Senate leaders have deteriorated after two years of draining battles over the budget. On Sunday, their antagonism produced a confusing day when talks seemed to be collapsing even as the two sides were moving closer to agreement on several fundamental issues.

Reid blamed McConnell for the impasse, saying Republicans were insisting on a change in the way inflation is measured that would serve to reduce Social Security benefits — a red flag for Democrats. Early in the day, Democratic aides described McConnell’s continued insistence on the change, known as “chained CPI,” as a “major setback.”

“At some point in the negotiating process, it becomes obvious when the other side is intentionally demanding concessions they know the other side’s not willing to make,” Reid said in a speech on the Senate floor.

McConnell countered by accusing Reid of “political gamesmanship” and announcing that he had telephoned Biden, opening up his own line of communication with the White House.

“I’m interested in a result here. And I’m willing to work with whomever can help,” McConnell said. “There is no single issue that remains an impossible sticking point. The sticking point appears to be a willingness, an interest, or courage to close the deal.”

Later, after consulting with fellow Republicans, McConnell agreed to take Social Security off the table.

The decision cheered Democrats. Obama had offered to include chained CPI in the big deficit-reduction package he had been negotiating with House Speaker John A. Boehner (R-Ohio) earlier this month. Obama endorsed the idea again Sunday, in an interview on NBC’s “Meet the Press,” calling it a tough decision he was willing to make “in pursuit of strengthening Social Security for the long term.”

But many Democrats say they would go along with the idea only as part of a far-reaching deal that also included at least $1.2 trillion in new tax revenue over the next decade. The deal under discussion Sunday would raise far less than that, somewhere between $600 billion and $800 billion.

“Chained CPI should be part of the larger discussion about the future of Social Security,” said Sen. Richard J. Durbin (D-Ill.), “not just a casual bargaining chip here at the last minute.”

Still, McConnell’s decision to drop the proposal served only to complicate the negotiations. By removing one of the few ideas on the table that would actually reduce spending, negotiators were left with a long list of new spending and only new taxes to cover the cost.

That infuriated many Republicans, who were already troubled by a deal that would require them to vote for a significant tax increase for the first time in more than 20 years. Republicans would be more open to the tax hike if it were used to reduce the federal budget deficit, but they remained resistant to raising taxes to expand government spending.

“You’re going to have to find some real spending cuts,” said Sen. Roy Blunt (R-Mo.). “You can’t just take this new tax revenue and say, ‘Well, we’ve got X number of dollars of new tax revenue, and that means we can spend that amount of money with no consequences.’ ”

In addition to delaying the sequester for two years — at a cost of roughly $200 billion — Democrats are seeking to protect doctors from a 27 percent cut in Medicare reimbursements set to hit in January, extend emergency benefits for the long-term unemployed for another year and approve a one-year extension of federal farm subsidies.

Democrats countered that Republicans, too, are in favor of much of the new spending. And in return, Democrats had made several concessions on taxes.

In addition to essentially agreeing to maintain the $5 million exemption and 35 percent tax rate above that value for inherited estates, Democrats also proposed to permanently protect middle-class households from the alternative minimum tax.

On the central issue of expiring tax cuts enacted during the George W. Bush administration, Democrats proposed to let the cuts expire on income over $450,000 a year for couples and $360,000 for individuals — a big jump from their original demand of $250,000 for couples and $200,000 for individuals.

But the Democrats insisted that tax rates on dividends and capital gains rise to 20 percent — from the current 15 percent — for couples and individuals above the lower thresholds. Those taxpayers would also lose some of the value of their personal exemptions and itemized deductions under the Democratic proposal.

Republicans were insisting late Sunday on even higher thresholds, arguing that income under $550,000 for couples and $450,000 for individuals should be completely protected.

As McConnell and Biden tried to bridge the divide, time had become as much of an enemy as the gritty details of tax and spending policy. Even if the leaders forge an agreement, the midnight deadline would be daunting to meet. Reid and McConnell would require the consent of all 100 senators to dispatch with the normal parliamentary procedures and complete debate and vote in hours rather than days.

And Senate passage would not guarantee an easy ride in the House, where Boehner’s conservative flank has shown deep resistance to any tax hikes. The speaker has indicated he does not want to approve a bill with mostly Democratic votes and a sliver of his 241-member Republican conference.

On Sunday, the House returned to work for the first time since the embarrassing failure of Boehner’s so-called “Plan B” option on Dec. 20. Republicans descended to a basement meeting room to eat pizza and receive a briefing from House leaders that was light on details.

Some lawmakers said they were anxious that the Senate would adopt a bill Monday and expect the House to act within a matter of hours. Instead, House members may push to delay a vote until Tuesday. That would mean missing the fiscal cliff deadline, but with U.S. financial markets closed for the New Year’s holiday, some members reasoned that the economy would be unlikely to suffer.

As House Republicans entered their meeting, aides filled the room with music, a Beatles classic. “Come together,” John Lennon sang, “right now.”

Rosalind S. Helderman contributed to this report.