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Sen. Lindsey Graham, South Carolina Republican, sounded even more pessimistic, predicting on CBS’ “Face the Nation” on Sunday that “I think we’re going over the cliff.” He also called the president’s proposed entitlement reforms “a joke.”

“The president’s plan does nothing but damn us to becoming Greece,” Mr. Graham said of the European country facing a budget crisis and 25.4 percent unemployment.

Mr. Geithner said Republican leaders in Congress are “in a very difficult place” because they are struggling to win support for a plan from their members, many of whom are committed to taking a hard-line stance against the president.

“We’re going to continue to look for ways to solve this problem, but ultimately [Republicans] have to come to us to say what they think they need,” the secretary said Sunday on ABC’s “This Week.” “What we can’t do is to keep guessing about what works for their caucus.”

Sen. Claire McCaskill, Missouri Democrat, told NBC’s “Meet the Press” on Sunday she “almost feel[s] sorry” for Mr. Boehner because “there is incredible pressure on him from a base of his party that is unreasonable about this.”

Mrs. McCaskill suggested her party hold firm because — if Republicans refuse to deal and the Bush tax cuts expire for everyone at year’s end — public pressure would force them to cave as soon as Congress returns to Washington in early January.

“They are going to get stuck with a raising of the rates of the top 2 percent either in a very painful way, or in a way that we can all suffer a little bit [and] address all three legs of the stool — entitlements, cuts and revenues — in a way that makes sense,” she said.

“It’s time for us all, maybe together, hold hands and say, let’s look at some of the things we’ve got to do structurally.”