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After the bitter brinkmanship that played out for weeks, the nation’s credit rating took a hit, and public approval ratings for all the top congressional leaders dropped, with Republicans taking a particularly hard beating.

But Mr. Norquist said Republicans shouldn’t be worried about approval ratings and argued that Republican losses in the House and Senate in the election had nothing to do with the debt-ceiling fight.

Before returning to Hawaii for a vacation this week, Mr. Obama warned Republicans against using a vote on the debt ceiling to try to win concessions on spending cuts. He insisted that he wouldn’t negotiate “with Congress over whether or not they should pay the bills they’ve already racked up through the law they have passed.”

Despite Mr. Obama’s line-in-the sand remarks, his former top budget adviser said the president has lost leverage for the budget battles to come and has no choice but to bargain with Republican lawmakers over raising the nation’s debt limit, which Congress will vote on in late February or early March.

“Well, I don’t know that [Mr. Obama’s] statement actually frankly matters that much,” Peter Orszag, the former director of the Office of Management and Budget, told CNBC late Wednesday. “You know, the sequester was only delayed by two months, so you’re going to have to negotiate over that. You’re going to have to negotiate over the appropriations bill for funding the daily operations of government.”

“So, if you’re negotiating over those things, and the debt limit’s kind of thrown in, you can say you’re not negotiating over the debt limit, but there will be negotiations in February or March.”

In the same interview, Mr. Orszag, now a vice chairman for global banking at Citigroup, said the White House lost bargaining power by not insisting that raising the debt limit be included in the fiscal cliff deal.

“My own perspective at least is I think the White House — in this second-best world — won that round, but they, by not insisting that the debt limit be tied to the package, it’s entirely possible they’re going to win the week and lose the quarter,” he said.