Sen. Tim ScottTimothy (Tim) Eugene ScottMcConnell names Senate GOP tax conferees GOP senator: Trump shouldn't pardon Flynn Trump should fill CFPB vacancy with Export-Import chief MORE (R-S.C.) refused to endorse Sen. Lindsey GrahamLindsey Olin GrahamGOP and Dems bitterly divided by immigration We are running out of time to protect Dreamers US trade deficit rises on record imports from China 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.