Sen. Tim ScottTimothy (Tim) Eugene ScottAfter Florida school shooting, vows for change but no clear path forward GOP senator: FBI failure in Florida shooting 'a separate issue' from Russia probe Sunday shows preview: Russian charges, Florida shooting dominate coverage MORE (R-S.C.) refused to endorse Sen. Lindsey GrahamLindsey Olin GrahamCongress punts fight over Dreamers to March Pence tours Rio Grande between US and Mexico GOP looks for Plan B after failure of immigration measures 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.