Republicans are even trying to turn concerns about their uncertain prospects in the presidential race into an opportunity in the battle for Senate supremacy, arguing that a Senate takeover could fundamentally alter the balance of power in Washington even if Mr. Obama won a second term. They are making the case to donors and voters that winning a Senate majority would give the party control of Congress and a united front against the president from Capitol Hill if Republicans hold the House.

It is not lost on Republicans that the last time a Democratic president won a second term, when Bill Clinton was re-elected in 1996, Senate Republicans still expanded their numbers…

Republicans say they see strong chances of victory in Montana over Mr. Tester and in Nebraska, where Senator Ben Nelson, a Democrat and a former governor, has trailed potential Republican rivals in polls.

“That leaves us 20 more seats to pick up one to get into the majority,” Mr. Cornyn said.