With the White House and congressional Republicans publicly deadlocked — but privately negotiating — to steer clear of the fiscal cliff, top lawmakers were left with little to do on Sunday’s talk shows except exchange partisan attacks, even as some predicted a deal would be reached.

In another sign that the parties were inching toward an agreement, Tennessee Sen. Bob Corker added his name to the list of Republicans who were acknowledging that an increase in tax rates on upper-income earners was on the horizon.

“There is a growing group of folks that are looking at this and realizing that we don’t have a lot of cards as it relates to the tax issue before year-end,” Corker said on “Fox News Sunday.”

If Republicans agree to extend the George W. Bush-era tax cuts for only families making less than $250,000, as sought by President Barack Obama, “the focus then shifts to entitlements, and maybe that puts us in a place where we actually can do something that really saves this nation,” Corker said. “I actually am beginning to believe that is the best route for us to take.”

Tax rates rise across the board with the coming of the new year, absent congressional action, and polls show the public would largely blame Republicans if no deal is reached.

“It’s not waving a white flag to recognize political reality,” Rep. Tom Cole, R-Okla., said on CNN’s “State of the Union.”

House GOP leaders, however, have yet to budge on tax rates. Majority Whip Kevin McCarthy of California said on NBC’s “Meet the Press” that Republicans had offered a proposal that raised $800 billion in revenue through curbing tax loopholes and deductions. He blasted Obama for campaigning across the country instead of sitting down at the table with Republicans.

“He’s a phone call away,” responded Senate Majority Whip Richard J. Durbin, D-Ill. A close Obama ally, Durbin said raising rates was the only way to secure enough revenue to stabilize the national debt. He said the GOP’s offer, like GOP presidential nominee Mitt Romney’s tax plans, just didn’t add up.

Still, there is a sense in Washington that an agreement is not as far off as the public sparring would suggest.

“I think we will get a deal. I think everyone realizes how important it is,” Sen. Charles E. Schumer, D-N.Y., said on “Fox News Sunday.’ “The reason I think we’ll get an agreement, what’s standing in the way is revenues, particularly making that top rate go up to 39.6 [percent]. But I think we are seeing real progress in that regard.”

Corker also agreed that a deal would be reached, and he said Republicans would soon have another chance to extract spending cuts.

“Republicans know that they have the debt ceiling right around the corner,” Corker said. “The leverage is going to shift.”

The Weekly Standard’s Bill Kristol, who has already suggested giving in on tax rates, said on Fox that Republicans should look to the debt ceiling as an opportunity, and he laid out a possible endgame of avoiding the cliff.

He suggested the GOP-controlled House pass two bills by the end of the year: one that would extend all of the tax cuts and another that would boost the top tax rate to 37 percent and be levied just on millionaires. He predicted the Senate would pass the second bill and Obama would sign it into law.

Former Sen. Alan Simpson, R-Wyo., and former White House Chief of Staff Erskine Bowles provided the comic relief for the ongoing drama.

Appearing on CBS’ “Face the Nation,” the two former chairmen of Obama’s 2010 blue ribbon deficit reduction commission implored the sides to get a deal for the good of the country. When host Bob Schieffer asked Simpson about his cutting a video where he implored young people to get involved that included some dancing Gangnam Style, Bowles could be heard cracking up off camera.

“Erskine, will you quit laughing,” Simpson said, snickering himself.

“It is great, isn’t it?” Schieffer said, unable to keep a straight face.

Simpson said his use of humor could be deployed just as much as sober analysis to strike an agreement.

“Nobody has any humor in Washington,” Simpson said.

Jason Dick contributed to this report.