WASHINGTON — The top Senate negotiators on the effort to prevent the government from going over the “fiscal cliff” offered a pessimistic assessment Sunday barely 24 hours before a deadline to avert tax hikes on virtually every worker.

Senate Republican leader Mitch McConnell said he’s yet to receive a response to an offer he made on Saturday evening to Majority Leader Harry Reid, D-Nev., the top Democratic negotiator. The Kentucky Republican said he’s reached out to Vice President Joe Biden in hopes of breaking the impasse and a McConnell spokesman confirmed the two have spoken.

Despite indications of progress in the negotiations, Democrats said Republicans were proposing to slow future cost of living increases for Social Security recipients as part of a compromise to avoid the cliff. Democrats rejected the idea.

“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.”

Reid said he has been trying to come up with a counteroffer but has been unable to do so. He said he’s been in frequent contact with President Barack Obama, who in a televised interview blamed Republicans for putting the nation’s shaky economy at risk.

“We have been talking to the Republicans ever since the election was over,” Obama said in the interview on NBC’s “Meet the Press” that aired Sunday. “They have had trouble saying yes to a number of repeated offers.”

The pessimistic turn came as the House and Senate returned to the Capitol for a rare Sunday session. The fate of the negotiations remained in doubt, two days before the beginning of a new year that would trigger across-the-board tax increases and spending cuts that leaders in both parties have said they want to avoid.

Reid said he is not “overly optimistic but I am cautiously optimistic” but reiterated that any agreement would not include the less generous inflation adjustment for Social Security.

“We’re willing to make difficult concessions as part of a balanced, comprehensive agreement, but we’ll not agree to cut Social Security benefits as part of a small or short-term agreement,” Reid said.

McConnell and Reid were hoping for a deal that would prevent higher taxes for most Americans while letting rates rise at higher income levels, although the precise point at which that would occur was a major sticking point.

Also at issue were the estate tax, taxes on investment income and dividends, continued benefits for the long-term unemployed and a pending 27.5 percent cut in payment levels for doctors who treat Medicare patients.

As the Senate opened a rare Sunday session, Chaplain Barry Black offered a timely prayer for lawmakers.

“Lord, show them the right thing to do and give them the courage to do it,” Black said. “Look with favor on our nation and save us from self-inflicted wounds.”

Senate negotiators were haggling over what threshold of income to set as the demarcation between current tax rates and higher tax rates. They were negotiating over estate limits and tax levels, how to extend unemployment benefits, how to prevent cuts in Medicare payments to doctors and how to keep a minimum income tax payment designed for the rich from hitting about 28 million middle class taxpayers.

Hopes for blocking across-the-board spending cuts were fading and Obama’s proposal to renew the two percentage point payroll tax cut wasn’t even part of the discussion.

Obama pressed lawmakers to start where both sides say they agree — sparing middle-class families from looming tax hikes.

“If we can get that done, that takes a big bite out of the fiscal cliff. It avoids the worst outcomes. And we’re then going to have some tough negotiations in terms of how we continue to reduce the deficit, grow the economy, create jobs,” Obama said in the NBC interview.

Gone, however, is the talk of a grand deal that would tackle broad spending and revenue demands and set the nation on a course to lower deficits. Obama and Republican House Speaker John Boehner were once a couple hundred billion dollars apart of a deal that would have reduced the deficit by more than $2 trillion over ten years.

Republicans have complained that Obama has demanded too much in tax revenue and hasn’t proposed sufficient cuts or savings in the nation’s massive health care programs.

Obama upped the pressure on Republicans to negotiate a fiscal deal, arguing that GOP leaders have rejected his past attempts to strike a bigger and more comprehensive bargain.

“The offers that I’ve made to them have been so fair that a lot of Democrats get mad at me,” Obama said.

Boehner disagreed, saying Sunday that the president had been unwilling to agree to anything “that would require him to stand up to his own party.”

Don Stewart, a spokesman for McConnell, said Sunday: “While the president was taping those discordant remarks yesterday, Sen. McConnell was in the office working to bring Republicans and Democrats together on a solution.”

The trimmed ambitions of today are a far cry from the upbeat bipartisan rhetoric of just six weeks ago, when the leadership of Congress went to the White House to set the stage for negotiations to come.

“I outlined a framework that deals with reforming our tax code and reforming our spending,” Boehner said as the leaders gathered on the White House driveway on Nov. 16.

“We understand that it has to be about cuts, it has to be about revenue, it has to be about growth, it has to be about the future,” House Democratic leader Nancy Pelosi said at the time. “I feel confident that a solution may be in sight.”

Sen. Dianne Feinstein, D-Calif., said the 2.1 million Americans whose extended unemployment benefits ran out on Saturday are already feeling the pain of Congress’ inaction.

“From this point on, it’s lose-lose. My big worry is a contraction of the economy, the loss of jobs, which could be well over 2 million in addition to the people already on unemployment. I think contraction of the economy would be just terrible for this nation. I think we need a deal, we should do a deal,” Feinstein said on “Fox News Sunday.”

But the deal was not meant to settle other outstanding issues, including more than $1 trillion in cuts over 10 years, divided equally between the Pentagon and other government spending. The deal also would not address an extension of the nation’s borrowing limit, which the government is on track to reach any day but which the Treasury can put off through accounting measures for about two months.

That means Obama and the Congress are already on a new collision path. Republicans say they intend to use the debt ceiling as leverage to extract more spending cuts from the president. Obama has been adamant that unlike 2011, when the country came close to defaulting on its debts, he will not yield to those Republican demands.

Lawmakers have until the new Congress convenes to pass any compromise, and even the calendar mattered. Democrats said they had been told House Republicans might reject a deal until after Jan. 1, to avoid a vote to raise taxes before they had technically gone up, and then vote to cut taxes after they had risen.

Associated Press writers David Espo, Julie Pace, Jim Kuhnhenn and Michele Salcedo contributed to this report.