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.