Sen. Tim ScottTimothy (Tim) Eugene ScottSenators introduce Trump-backed criminal justice bill Time to pass the First Step Act Debbie Stabenow reelected to a fourth Senate term in Michigan MORE (R-S.C.) refused to endorse Sen. Lindsey GrahamLindsey Olin GrahamOn The Money: Why the tax law failed to save the GOP majority | Grassley opts for Finance gavel, setting Graham up for Judiciary | Trump says China eager for trade deal | Facebook reeling after damning NYT report Cotton opposes Trump-backed criminal justice bill Lawmakers say California will eventually get emergency funding for fire relief MORE (R-S.C.), telling CNN he's focused on his own race and not his fellow senator's primary challenge.

"I am up for reelection next year myself. I'm going to make sure that Tim Scott gets out … and going to allow for all the other folks on the ballot to represent themselves very well, but I'm going to work hard for my election," he said when asked on CNN's Crossfire if he'd back Graham.

"I'm certainly going to work really hard for Tim Scott — got to win first," he said when asked again.

Graham, who has broken with the GOP base on issues like immigration and climate change, is facing a number of Tea Party challengers, though he appears to be in strong shape for reelection. Scott, who has a more conservative reputation, was appointed to the Senate early this year and faces his own reelection, though he doesn't have any serious competition.