But Democrats and Republicans are largely resisting his call. A bipartisan measure in the Senate has gained no support from members other than the two who wrote it, and a more conservative version offered by Senator Lindsey Graham of South Carolina has only a handful of Republican backers.

Republicans are loath to give Mr. Obama more authority because they are highly critical of his strategy in Syria, and Democrats are still stinging from the vote in 2002 to authorize the war in Iraq. Even among supporters, there are partisan divisions, with Democrats preferring a measure that would limit the president’s authority and Republicans preferring to allow a more robust use of force.

Members of both parties, especially the Republicans who control Congress, fear that a high-profile debate followed by a failed vote to authorize force would be a disastrous public display of division, perhaps emboldening enemies abroad. Lawmakers and Mr. Obama were embarrassed in 2013 when Congress did not authorize airstrikes against President Bashar al-Assad of Syria.

“There are not many things that we do that are more important,” said Senator Jeff Flake, Republican of Arizona, who has been pushing for a measure for months with Senator Tim Kaine, Democrat of Virginia. Mr. Flake said that most members of Congress simply did not want to take a tough vote. “That is the most accurate explanation there is,” he said. “It’s unfortunate, but I think that’s it.”

Mr. Flake and others believe that after the recent attacks in Paris and in San Bernardino, Calif., and with the new push from Mr. Obama, Congress may feel pressure to step up in the new year.

“I think it’s a very good idea,” said Senator Chuck Schumer, Democrat of New York. “All we’ve been hearing from the Republican side is that the president is not doing enough, so an A.U.M.F. forces Republicans to outline their proposals.”