WASHINGTON – A deeply divided Congress on Sunday pushed the nation perilously closer to a Jan. 1 “fiscal cliff” of record tax hikes and draconian spending cuts, as late involvement by Vice President Biden did nothing to resolve a partisan stalemate on the issue.

A day of anxious waiting on Capitol Hill ended shortly before 6 p.m., when Senate Majority Leader Harry Reid announced that the Senate’s rare Sunday session was ending.

“There’s still time left to reach an agreement, and we intend to continue negotiations,” said Reid, D-Nevada, who also acknowledged that there is “still significant distance between the two sides.”

That being the case, other lawmakers were pessimistic.

“I’m incredibly disappointed we cannot seem to find common ground,” Sen. Lindsey Graham, R-S.C., tweeted earlier in the afternoon. “I think we’re going over the cliff.”

Reed adjourned the Senate until 11 a.m. today, when it will begin its first New Year’s Eve session in 42 years. Meanwhile, the House conducted a series of votes on noncontroversial measures as it awaited the Senate to take the lead in resolving the fiscal cliff.

Hopes that a deal might be struck were raised briefly Sunday when Senate Minority Leader Mitch McConnell, R-Ky., told colleagues that he had been in telephone contact with Biden.

Biden and McConnell are experienced in crafting the sort of last-minute, “Perils-of-Pauline” deals that have become the only way to get anything major done on Capitol Hill.

In late 2010, for example, they collaborated on a two-year extension of the George W. Bush-era tax cuts, and in summer 2011 they worked on the deal to raise the federal debt ceiling in exchange for a promise of future budget cuts.

But those two deals combined to create the fiscal cliff, setting a Jan. 1, 2013, deadline for when those tax cuts are set to expire – thereby raising taxes on more than 90 percent of the American people – and when a set of deep spending cuts, equally split between defense and domestic programs, are set to take effect.

Federal Reserve Chairman Ben Bernanke has warned that those tax increases and spending cuts could push the nation back into recession.

But that has done nothing to narrow the chasm between Democrats insisting on higher taxes on the wealthy and Republicans who are reluctant to increase any tax, ever.

President Obama remade his case for compromise in an appearance Sunday on NBC’s “Meet the Press.”

“We have been talking to the Republicans ever since the election was over. They have had trouble saying yes to a number of repeated offers,” Obama said.

Hearing that, House Speaker John Boehner, R-Ohio, fired back at the president in a written statement.

“Americans elected President Obama to lead, not cast blame,” Boehner said.

Behind the scenes, it seemed, negotiations were every bit as acrimonious as the comments by Obama and Boehner.

Reid and McConnell’s negotiations ground to a halt early Sunday when Republicans demanded that the deal include a new measure of inflation that would be used to calculate Social Security cost-of-living increases.

In effect, that change would mean lower raises for Social Security recipients – something Reid and other Democrats refused to agree to, and that perhaps even raised concerns among Republicans. Emerging from a mid-afternoon meeting, several GOP senators said the move to change the inflation measure had been abandoned.

Hopes for such a big deal have long ago evaporated, and now lawmakers are focusing on the possibility of a much narrower deal: one that would prevent most of the tax hikes that are set to take effect this week, while allowing the automatic spending cuts to go through, at least for the time being.

Obama has insisted that taxes be raised on families earning more than $250,000, but he’s also shown a willingness to settle for a threshold of $400,000.

That being the case, aides to Reid and McConnell were engaged in a negotiation that could end with a deal raising taxes on those making more than $400,000, congressional aides said Sunday. Democrats were pushing a proposal to raise taxes on those making more than $360,000, while Republicans were holding out for a hike only for those making more than $450,000.

The two sides also appeared to remain divided over the estate tax. Democrats want to raise the estate tax rate from the current 35 percent to 45 percent while taxing all estates worth more than $3.5 million, up from the current threshold of $5.2 million per person. But Republicans were resisting any changes to the estate tax.

With negotiations continuing, lawmakers from both parties said they were worried about the lack of progress.

“Our membership is as frustrated as the American people we were sworn to serve,” said Rep. John Larson of Connecticut, chairman of the House Democratic Caucus.

“I’m concerned with the lack of urgency here. There’s far too much at stake,” 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.”

Of course, that is nothing new in Washington, where Congress has almost allowed the government to shut down and the federal treasury to max out its debt limit before striking last-minute budget deals in recent years.

But there is one difference this time: Now Congress will spend New Year’s Eve trying to avert a crisis it concocted.

Which is why David Frum, a former speech writer for former President George W. Bush, tweeted: “As we head toward New Year’s Eve deadline, interesting possibility that Congress will determine fiscal future of U.S. while intoxicated.”

News wire services contributed to this report email: