Sen. Lindsey GrahamLindsey Olin GrahamThe Memo: Trump can't let go of McCain grudge The Hill's Morning Report - Trump's intraparty feuds divide Republicans Trump's attacks on McCain exacerbate tensions with Senate GOP MORE (S.C.) blasted fellow Republican presidential candidate Donald TrumpDonald John TrumpSenate GOP budget ignores Trump, cuts defense Trump says he'll nominate Stephen Moore to Fed White House: ISIS territory in Syria has been 100 percent eliminated MORE on Monday night, calling the businessman a "jackass" for his comments on Sen. John McCainJohn Sidney McCainLou Dobbs: Political criticism of McCain 'not an exhumation of his body' Trump rips McCain, says he gave Steele dossier to FBI for 'very evil purposes' The Hill's Morning Report — Washington readies for Mueller end game 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.