Sen. Tim ScottTimothy (Tim) Eugene ScottIt's time to empower military families with education freedom GOP Sen. Tim Scott says if he runs in 2022 it will be his last race When it comes to student debt, it is time to talk solutions MORE (R-S.C.) refused to endorse Sen. Lindsey GrahamLindsey Olin GrahamWhat would John McCain do? Sunday shows preview: Trump ratchets up trade war with China White House won't move forward with billions in foreign aid cuts 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.