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Treading down that path could provide more flexibility to negotiators on both sides, but it also would leave Congress without a clear vehicle with which to move the extension. Budget negotiators have created a few self-imposed deadlines, but benefits for the 1.3 million people will end Dec. 28 absent action from Congress.

In December 2010, Mr. Obama and congressional Democrats traded a two-year extension of the George W. Bush-era tax cuts for a $57 billion, one-year extension of emergency unemployment benefits.

But with Democrats and Republicans seemingly unwilling to budge on major entitlement reforms and tax increases, respectively, it’s unclear what Republicans could receive in a possible deal for an extension.

Rep. Chris Van Hollen of Maryland, the ranking Democrat on the House Budget Committee, said the program could be funded by squeezing additional revenue out of “deadbeat” taxpayers and Americans trying to hide money overseas.

“We do pay for it,” he said. “We pay for it by making sure that people who are deadbeats and trying to hide their money pay their fair share of taxes so that other people don’t have to pay more.”

But a philosophical divide on the issue between the two parties was also evident Sunday. Sen. Rand Paul, Kentucky Republican, said he supports unemployment benefits for the 26 weeks that are paid for but extending them beyond that would be a disservice to workers.

“When you allow people to be on unemployment insurance for 99 weeks, you’re causing them to become part of this perpetual unemployed group in our economy, and while it seems good, it actually does a disservice to the people you’re trying to help,” he said.