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“That D.C. insiders on a bipartisan basis think highly enough of Biden’s political talents and credibility that they want him involved at a time when the nation is perilously close to the fiscal cliff, that’s an impressive recognition,” said Joel Goldstein, a vice presidential scholar and a professor at St. Louis University School of Law. “This is truly a historic position for a vice president to be called upon to fulfill.”

Even though he knows Mr. Biden well, Sen. Benjamin L. Cardin, Maryland Democrat, said the vice president never talks to him about plans to run for president in 2016.

When asked whether the vice president’s key role in jump-starting talks this week would improve his presidential prospects, Mr. Cardin said only that he would like to see the finer points of any debt-reduction package.

“We’ll have to see the details of the deal,” Mr. Cardin said.

When asked whether Mr. Biden’s last-minute role in the negotiations could strengthen a future White House bid, Sen. Patrick J. Leahy, Vermont Democrat and the longest-serving senator, said the vice president is focused on helping prevent an economic crisis for the country, not his political future.

“Knowing Joe, he’s more concerned right now about getting a deal for the American people,” Mr. Leahy said.