Graham defended McCain's role and said that he told Trump that his close friend's involvement was limited.
Graham said that he "was very direct" with Trump, and if Trump believes it was the McCain people who spread this,"he's right. There were some McCain people that took a piece of garbage and tried to go after Trump after the election."
Graham added: "I told the President it was not John McCain. I know because John McCain showed me the dossier. And I told him the only thing I knew to do with it, it could be a bunch of garbage, it could be true, who knows? Turn it over to somebody who's job it is to find these things out and John McCain acted appropriately."
Asked more about the matter later, Graham said he put the dossier in "a safe" the day he got it and turned it over to the FBI the next day.
"And I understand that, clearly people are in the McCain world that did some things inappropriate but it was not John McCain. John McCain did not give it to anybody in the press, he talked to me just as soon as he got it, and he turned it over to the FBI and that's exactly what he should have done.
Former McCain aide David Kramer approached the Arizona senator after the 2016 elections with former British spy Christopher Steele's dossier of Trump-Russia connections, and then McCain said he delivered the documents to the FBI, saying that was the extent of his involvement.
In November 2016, a former British ambassador to Russia who knew about what Steele had researched met McCain, his staffer Christopher Brose and Kramer, a former State Department official who worked alongside McCain, in Halifax.
At McCain's request, Kramer followed up with Steele, then received copies of the dossier from an owner of Fusion GPS.
Kramer then shared the dossier with McCain and the Senate staffer on November 30, 2016, and asked McCain to pass it to the FBI and CIA.
On Monday, Graham said: "You know, I don't care if President Trump likes John McCain or not, I do. I like President Trump. John McCain is one of my closest friends in life, not just in politics, I want President Trump to be successful. He's been very good to me in the sense he's let me in his world. And I have some access to the President about things that matter to me and the country and I appreciate that, and I want him to be successful."
Graham said McCain is "one of his best friends in life." Graham said Trump asked the South Carolina senator: "He really was your friend?"
"I said, 'yeah, he really was my friend,'" Graham said.
This story has been updated with additional developments Monday.
CNN's Katelyn Polantz contributed to this report.