Sen. Lindsey Graham said in an interview Thursday that he'll do "everything he can" to help President Donald Trump, but at the same time, the comments the president has made about the late Sen. John McCain bothered him "greatly."

"It pisses me off to no end, and I let the president know it," the South Carolina Republican, who had shared a close friendship with McCain, told "CBS This Morning" co-host John Dickerson.

He also said he found Trump's response to McCain's death "disturbing," after the president ignored questions from reporters about the death and after the decision to put the White House flag back at full-mast shortly after the senator's death, and then to pull it back down again.

Graham said he did not call Trump directly to get him to bring the flag back down, but he did call "some people around him."

On Wednesday, Graham told CNN that Trump had called him after his speech on the Senate floor about McCain and "couldn't have been nicer."

Meanwhile, there have been calls to rename the Russell Senate Office Building after the Arizona senator, but Graham said if he could pass some legislation instead to honor his friend, it would be to reform immigration.

"I want to get that done because it meant a lot to John to fix a broken immigration system," said Graham. "The president can do this. He is a dealmaker. But the people around him, not so much."

He also said he believes there is much the president could learn from McCain.

"What I would tell the president, you've got a lot of people you think are treating you unfairly," said Graham. "Fight back. But you're going to have to be a big man in a big office. John McCain was a big man, worthy of a big country. Mr. President, you need to be the big man that the presidency requires."