"We can't have a nominee be an albatross around the down-ballot races," Cornyn told CNN when asked if he had concerns about the prospect of Trump winning the GOP race. "That's a concern of mine."
When asked if Trump would be a drain on the party's bid to keep control of the Senate, the Texas senator said, "I think he certainly is a controversial figure. I think we need someone who can unify the party, as opposed to divide the party."
The Republican's comments came a day before his own state votes on Super Tuesday, when 595 delegates will be at stake in states from Georgia to Alaska. Trump is poised to rack up a huge victory, potentially setting him on a clear path towards locking up the GOP nomination.
But the businessman's penchant for making off-color and stunning remarks -- the latest when he initially refused to repudiate David Duke on CNN's "State of the Union with Jake Tapper" -- have made top Republicans downright fearful that he could be detrimental to their hopes of retaking the White House, keeping the Senate and effectively filling the ninth spot on the Supreme Court.
Sen. Ben Sasse, a freshman Republican from Nebraska, announced Sunday night he would not support Trump if he became the nominee. And a number of Senate Republicans in swing states could be in a difficult spot to unify behind Trump given the more moderate leanings of their respective electorates.
Cornyn suggested that if Trump were the nominee some of his incumbents in Senate races -- where 24 GOP seats are in contention, compared to 10 for the Democrats -- could have to distance themselves from the debate that's happening nationally and on the campaign trail.
"It's always important for candidates, in my experience, to be in tune with your state and not necessarily what's happening in Washington," said Cornyn, who has not publicly endorsed any candidate in the race.
Both Senate Majority Leader Mitch McConnell and House Speaker Paul Ryan have said little publicly about the presidential race. But in his most extensive public remarks last December, McConnell expressed hope that GOP voters would nominate a candidate who could succeed in purple states, several of which have heavily contested Senate races.
"Unless the nominee for President can carry purple states, he's not going to get elected," McConnell told reporters at the time.