Sen. Lindsey GrahamLindsey GrahamSenate takes up NATO membership for Montenegro Republicans giving Univision the cold shoulder: report Graham: 'I'm glad' Ivanka will be working in the White House MORE (S.C.) blasted fellow Republican presidential candidate Donald TrumpDonald TrumpTrump wants B for 62 miles of border wall: report Trump can boost national security, own legacy with Russian arms deal Press: Did Trump learn? MORE on Monday night, calling the businessman a "jackass" for his comments on Sen. John McCainJohn McCainSenate takes up NATO membership for Montenegro A great military requires greater spending than Trump has proposed Cheney: Russian election interference could be ‘act of war’ MORE's (R-Ariz.) war record.

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"What he said about John, I think, was offensive. He's becoming a jackass at a time when we need to have a serious debate about the future of the party and the country," Graham said on CNN’s "Erin Burnett OutFront."

"This is a line he's crossed, and this is the beginning of the end of Donald Trump. ... I am really pissed," Graham added.

Graham repeated his criticism of Trump on Tuesday morning, saying he didn't think there was "a market for slandering POWs" in the Republican Party.

"I don’t care if he drops out," Graham said on "CBS This Morning," after a top Iowa newspaper called on Trump to drop out of the race in the wake of his remarks.

"Stay in the race, just stop being a jackass. You don’t have to run for president and be the world’s biggest jackass — that’s not your choices," Graham said on CBS.

Trump caught widespread flak over the weekend for questioning McCain's status as a war hero, saying that he preferred those who did not get captured. He has since attempted to clarify those remarks.

McCain, who was held as a prisoner of war during Vietnam for nearly six years, brushed off the comments earlier Monday, saying Trump might owe an apology to other POWs.

Graham, a close friend of McCain's, is a long shot for the White House in an extremely crowded GOP field, but has sought to make national security a primary focus of the 2016 race.

Updated at 10:16 a.m.