A small but growing group of Republicans say the party should perhaps accede to President Obamas demand for higher tax rates for top earners so that the attention can shift to making serious cuts in benefit programs like Medicare and Medicaid, a top Republican senator said on Sunday. The senator, Bob Corker of Tennessee, a member of the Banking Committee who had earlier presented a deficit-reduction plan of his own, said on Fox News Sunday that if Republicans gave in to the presidents chief demand, then all of a sudden, the shift goes back to entitlements and maybe it puts us in a place where we actually can do something that really saves the nation.
Within hours of Mr. Corkers comments, Mr. Obama and Speaker John A. Boehner met privately at the White House for negotiations. Administration officials would not offer details of the discussion.
The disclosure of the meeting, however, indicated that private discussions continue in the face of Republican leaders public statements decrying the lack of progress and the presidents refusal so far to specify the sort of deep, long-term reductions in spending for Medicare and other social programs that they insist upon as a condition of their support for raising taxes from high earners.
The White House and Mr. Boehners office issued identical statements afterward saying, The lines of communication remain open. On Friday, Mr. Boehner told reporters that another week had been wasted, with just three weeks left for lawmakers to avert a fiscal crisis.
Also on Friday, Mr. Obama spoke privately with Congressional Democratic leaders, Senator Harry Reid, the Senate majority leader, and Representative Nancy Pelosi, the House minority leader, presumably to gauge what positions the Democrats in Congress could support.
Republicans have been insisting that the Obama administration agree to substantial savings in entitlement programs as the two sides negotiate how to narrow the countrys huge deficits. But Mr. Corker is part of a group of Republicans who say that the party ultimately will have to yield to the Obama demand of higher tax rates for top earners, potentially back to the levels that prevailed under President Bill Clinton.
That group of Republicans, he said on Fox, was beginning to realize that we dont have a lot of cards as it relates to the tax issue before year end but that a tax-rate concession could be converted, jiu-jitsu style, into a tactical advantage.
Mr. Corkers comments came as key figures on network news programs mixed cautious words of optimism over the fiscal-crisis negotiations between the White House and Congressional Republicans with dire warnings about what a failure to act might bring.
Christine Lagarde, the managing director of the International Monetary Fund, warned that such a failure was the gravest threat now facing the still-fragile United States economy greater than the European debt crisis or any uncertainty in China.
It would result in the stock market really taking a hit, Ms. Lagarde said on the CNN program State of the Union. The overall result could be zero growth next year, not the 2.1 percent growth rate projected by the I.M.F.
She called for a balanced agreement of both revenue increases and spending cuts. That is a word President Obama has often used to describe his preferred outcome. Ms. Lagarde offered a tentative prediction that what she called American pragmatism would prevail and avert the worst outcome.
Senator Charles E. Schumer of New York, the No. 3 Senate Democrat, also predicted that an agreement would be reached. Perhaps more optimistically, he said he thought it would include a stipulation that the national debt ceiling be raised, without further drama, when it comes due in late January or February.
I believe, frankly, our Republican colleagues have learned that to say the government is not going to pay its debts and hold it up for something else is bad substance and bad politics, he said on Fox News Sunday. I dont think theyll prevail on that.
Mr. Corker said he agreed with Mr. Schumer that a deal would be reached and were not going to go over the fiscal cliff. But he suggested that Republicans would not so easily give up on the debt-ceiling leverage. Republicans know that they have the debt ceiling thats coming up right around the corner and the leverage is going to shift, he said.
A Senate vote is required to raise the nations statutory borrowing limit.
One author of the much-disputed Simpson-Bowles debt-reduction plan, the former Wyoming senator Alan Simpson, a Republican, expressed some optimism on Sunday. He said he thought a deal would be reached but that if it were too narrow if it failed to make deep enough cuts in health care spending or other big social programs the economy, and Americans, would still suffer.
Im an optimist, he said on the CBS program Face the Nation. I think theyll do something, but if they do small ball, that wont work: The markets arent going to listen to that.
His partner in the plan, Erskine Bowles, who was White House chief of staff under President Bill Clinton, warned that a failure in the ongoing negotiations would be disastrous. But he said he was somewhat more upbeat than he was a week ago, because of signs of movement from both sides.
After weeks of what he called kabuki theater, the two sides have started to tango now, Mr. Bowles said, also on CBS. Asked if a deal could be reached by the end of the year, Mr. Bowles replied, They absolutely can do it. If they dont do it, shame on them.
Mr. Corker, who will be the ranking Republican on the Foreign Relations Committee in the new Congress, was also asked whether he thought Mr. Obama should nominate Susan Rice, the United States ambassador to the United Nations, to succeed Hillary Rodham Clinton as secretary of state, given criticism of Ms. Rices role in the Sept. 11 attack on the American diplomatic mission in Benghazi, Libya.
I do think that Ambassador Rice is viewed as a political operative, and I dont think hes going to nominate her, I really dont, Mr. Corker said. I think that time has come and gone. He said that if she was nominated, he would give her a fair hearing, but that Ms. Rice would probably face opposition from both sides of the aisle.
Two Democratic senators disagreed, however. Senator Richard J. Durbin of Illinois, the No. 2 Democrat in the Senate, said he believed Ms. Rice would be confirmed. I do believe so, he said on the NBC program Meet the Press. Shes an extraordinary person.
And Mr. Schumer suggested a way for Republicans to express opposition without taking the drastic step of blocking what would be one of Mr. Obamas most important second- term nominations.
If shes nominated by the president and I think shes very capable they shouldnt filibuster, he said. They dont have to vote for her, but dont require 60 votes to get her on, and thats traditionally whats been done.