Senate Majority Leader Harry Reid, D-Nev., said the negotiations were still going on so reaching a deal was still possible, The Washington Post reported.
"There's still time left to reach an agreement, and we intend to continue negotiations," Reid said on the Senate floor.
Sen. Mike Johanns, R-Neb., said the Republican and Democratic negotiators were "so close that they can't afford to walk away."
Sen. Lindsey Graham, R-S.C., wasn't so optimistic.
"I think we're going over the cliff," he said.
Sen. Kay Bailey Hutchison was blunt.
"It just looks like we can't govern," the retiring Texas Republican said.
The Senate was to be back in session at 11 a.m. Monday, the Post noted.
Direct talks on the fiscal cliff began Sunday between Senate Minority Leader Mitch McConnell and Vice President Biden, Republicans said
The Post said Republicans came out of a caucus meeting saying McConnell had taken a call from the vice president.
Senate Republicans also said they gave way on a demand to cut Social Security payments.
The New York Times reported Republican senators withdrew a demand that any deal include a new way of calculating inflation. The new method would have lowered payments to programs such as Social Security, plus slow their growth, the newspaper said.
Republicans met behind close doors, and then said they were withdrawing the demand, which had brought the negotiations to a deadlock.
Sen. John McCain, R-Ariz., said protecting high-income households from tax increases while demanding Social Security cuts was "not a winning hand," the Times reported.
Earlier, McConnell called on Biden for help to get the stalled fiscal cliff talks going again.
On the Senate floor, McConnell said he asked the White House for help shortly after Democratic aides said talks between McConnell, R-Ky., and Reid had struck a "major setback," the Post reported. In turn, Reid said he withdrawing from the talks to allow McConnell and Biden to negotiate.
McConnell said he has received no Democratic response to an offer he made Saturday evening.
The minority leader said, "The sticking point appears to be the willingness and interest or frankly the courage to close the deal. I want everyone to know, I'm willing to get this done, but I need a dance partner."
Reid answered quickly he had spoken to President Obama several times Sunday morning, the Post reported, but could not provide a counter-offer. Fox News reported Senate Democrats said the talks stalled after Senate Republicans proposed a deal that included the changes on how Social Security payments are calculated.
Meanwhile, Politico reported Biden returned to the White House Sunday from his home in Delaware. A Biden aide refused to say why.
Bush-era tax cuts, which have been extended several times, are due to expire Jan. 1 at the same time across-the-board spending cuts imposed by the Budget Control Act of 2011 take effect -- pushing the economy over the so-called fiscal cliff economists say could throw the U.S. economy into recession.
Both sides in the fiscal cliff debate predicted no one would be entirely satisfied by any deal the U.S. Congress hammers out in Washington.
Congress resumed debate on the thorny issue Sunday with no clear signs the Republicans or Democrats were prepared to give ground on tax hikes and cuts to entitlement spending.
Graham said on "Fox News Sunday" Congress was aware the window of opportunity was closing and some type of agreement, probably a short-term fix, was more likely.
"I think the leadership now is working," Graham said. "It will probably be an immediate deal. In other words, it's not going to solve all of the problems forever, but it is going to solve certain problems."
Sen. Dianne Feinstein, D-Calif., indicated the Democrats were more prepared to accept additional spending cuts as a painful alternative to the more-Draconian sequestration plan; however she reiterated that cutting taxes further would only put the United States deeper in the hole.
"I think there is a commitment to cut spending," Feinstein told Fox. "[Because] if you do it by virtue of sequestration it falls, regardless of priority, on things that are very high priority that should not be cut."
Other members of Congress expressed hope Sunday a deal could be hammered out, but said the devil was still firmly in the details.
Appearing on ABC's "This Week," senior senators and House members agreed a fair amount of political posturing was taking place, but that didn't discount the fact there remained significant differences between Democrats and Republicans.
"Until we know exactly what the Senate's proposing, the senators couldn't tell us the details, it's impossible to know," said Rep. Chris Van Hollen, D-Md. "It depends on the deal."
Meanwhile, Obama said finalizing a deal had boiled down to the question of tax increases for wealthy Americans.
The president said on NBC's "Meet the Press" it was the Republicans who were holding up an agreement by digging in on the tax-hike question.
"They say that their biggest priority is making sure that we deal with the deficit in a serious way, but the way they're behaving is that their only priority is making sure that tax breaks for the wealthiest Americans are protected," Obama said in the interview that was taped Saturday.
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