"I don't think he's racist but he's playing the race card. And in the political process he's putting the race card on the table," he told CNN Tuesday. "I think it's very un-American for a political leader to question whether a person can judge based on his heritage."
"If he continues this line of attack then I think people really need to reconsider the future of the party," Graham added.
The senator also urged Republicans backing Trump to rescind their endorsements following the presumptive Republican nominee's comments about judges' ethnicity and religion and discomfort with a potential jurist of the Muslim faith.
"This is the most un-American thing from a politician since Joe McCarthy," Graham told
The New York Times.
"If anybody was looking for an off-ramp, this is probably it," he added. "There'll come a time when the love of country will trump hatred of Hillary" Clinton.
Trump is facing scrutiny for comments he made over the past week accusing Muslim and Mexican judges of being biased. He was asked Sunday on CBS's "Face the Nation" if a Muslim judge would treat him unfairly due to his proposal to ban Muslims from entering the United States.
"It's possible, yes," Trump said. "Yes. That would be possible, absolutely."
The businessman was already under fire for his assertion that the Mexican parentage of Judge Gonzalo Curiel has colored his rulings in lawsuits against Trump's business school, given Trump's plan to build a wall on the U.S.-Mexico border.
CNN has reached out to the Trump campaign for comments.
Trump surrogate New Jersey Gov. Chris Christie dismissed Graham's comments as having any significance.
"Lindsey's lost any credibility he's had. He should worry about going back to South Carolina and trying to rebuild his base in South Carolina or he won't be in the United States Senate for much longer," he said Tuesday.
Although Graham previously told CNN that he is supporting neither Trump nor Clinton, he did say that a phone call with Trump in May was a "cordial, pleasant phone conversation" and said the two discussed national security threats, including ISIS. Graham also told CNN's Dana Bash
in May that he informed donors that he will no longer pick a fight with Trump because it does no good.
Graham and Trump have had an acrimonious relationship stretching back to some of the earliest days of the Republican presidential race, when Graham was also a Republican primary candidate.
One of their most notable beefs occurred last summer when Trump released
Graham's personal cell phone number during a press conference.
The senator, who dropped his presidential bid in December, has since offered some of the most biting criticisms of Trump, including calling him a "race-baiting, xenophobic, religious bigot" late last year.
And after Trump locked up the Republican nomination, Graham said that Trump doesn't have the temperament or judgment to be president and said Trump has "conned" the Republican Party.
'He's going to move into a very different place'
Tennessee Republican Sen. Bob Corker, who is potentially on Trump's running mate short list , repeatedly told reporters he didn't condone Trump's comments on Curiel and said the Trump campaign has two to three weeks "to redirect their efforts in a positive way."
"I'm hopeful that over the course of the next couple of weeks -- in this very important time in his campaign -- that he's going to move into a very different place. He's got an opportunity that anybody who serves in public service would love to have and that is to really be a positive force in changing our trajectory and I hope he won't let that opportunity slip through his hands by continuing some of the things that have occurred over the last five to six days."
Corker said he's weighed in with the campaign said the debate is "just not a place for our country to be."
"We need to move beyond personality type issues based issues to substantive policy issues," Corker told reporters before heading into a party luncheon.
Corker, chairman of the Senate Foreign Policy Committee, said that he spoke to Trump after his foreign policy speech, which he thought was "a step away" from the personality discussion, and he and other Republicans hoped the GOP nominee would shift gears.
"I couldn't agree more that the last five to six days has been very negative, yet at that same time I understand that I know people's hair is on fire and very concerned. Look I don't condone what has occurred."