By Veronica Stracqualursi, CNN

(CNN) -- Sen. Lindsey Graham said Tuesday that President Donald Trump's relationship with Jeff Sessions is "beyond repair" and suggested that the attorney general be replaced.

"The President has lost confidence in Jeff Sessions. I'm telling you what everybody in the country knows, this is a dysfunctional relationship, we need a better one," the South Carolina Republican said in an interview on NBC News' "Today" show.

Graham defended Sessions' decision to recuse himself from the Justice Department's investigation into Russian election interference due to his involvement with the Trump campaign.

"But we need an attorney general that can work with the President, that can lead the Department of Justice. This relationship is beyond repair, I think," Graham said. "I'm not asking him to be fired, but the relationship is not working."

Graham suggested that Sessions needs to be replaced with "somebody who is highly qualified and will commit to the Senate to allow" special counsel Robert Mueller to "do his job."

"Is there somebody who's highly qualified that has the confidence of the President, will also understand their job is to protect Mueller? Yes, I think we can find that person after the election if that's what the President wants," Graham argued.

The South Carolina Republican appeared to suggest that there's another reason Trump has been criticizing Sessions other than not ending Mueller's investigation.

"It's much deeper than that," Graham told NBC News.

"We won't say on this show, but it's a pretty deep breach," he said, without elaborating.

The senator's comments come after Trump's interview with Fox News last week in which the President rebuked Sessions' decision to recuse himself from the Russia investigation and again disparaged Sessions as ineffective, saying he "never took control of the Department of Justice."

Sessions hit back in a rare statement saying, "While I am Attorney General, the actions of the Department of Justice will not be improperly influenced by political considerations."

Last week, Graham became one of the first Republicans on the Senate Judiciary Committee to signal an openness to Trump ousting Sessions in his first term when he said the President deserved an attorney general he had "faith in."