A small but growing group of Republicans say the party should perhaps accede to President Barack Obama’s 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 Sunday.

Sen. 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 president’s 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 Corker’s comments, Obama and Speaker John 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 president’s 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 Boehner’s office issued identical statements afterward saying, “The lines of communication remain open.” On Friday, Boehner told reporters that another week had been wasted, with just three weeks left for lawmakers to avert a fiscal crisis.

Also on Friday, Obama spoke privately with congressional Democratic leaders, Sen. Harry Reid, the Senate majority leader, and Rep. 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 country’s huge deficits. But 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 don’t have a lot of cards as it relates to the tax issue before year-end” — but that a tax-rate concession could be converted into a tactical advantage.

Corker’s 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 U.S. economy — greater than the European debt crisis or any uncertainty in China.

“It would result in the stock market really taking a hit,” 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 IMF.