Speaking at a news conference in Trump Tower in New York on Tuesday, Trump repeatedly called Kristol a "loser." He said The Weekly Standard publication was "failing" and questioned Kristol's political judgment.
"He's been wrong for two years," Trump said.
Kristol soon fired back on Twitter.
"I gather Donald Trump said I'm a loser. I've won some and I've lost some, but one thing I've always tried not to be is a roaring jackass," Kristol tweeted Tuesday afternoon.
Trump criticized Kristol in a series of tweets Sunday night, calling him a "dummy" and "an embarrassed loser," suggesting the pundit reflects a lack of party unity that could affect the ideological composition of the Supreme Court.
"The Republican Party has to be smart & strong if it wants to win in November. Can't allow lightweights to set up a spoiler Indie candidate!" Trump tweeted.
Kristol did not give any indication of who the candidate might be, other than saying the person will have a "strong team and a real chance."
"Just a heads up over this holiday weekend: There will be an independent candidate--an impressive one, with a strong team and a real chance," Kristol tweeted on Sunday.
Reached for comment Monday morning, Kristol would only say, "Still lots of I's to dot and T's to cross."
A persistent critic of the presumptive GOP nominee, Kristol has repeatedly been the center of rumblings and machinations around a possible independent bid.
Earlier in May, Kristol met with
former Republican presidential nominee Mitt Romney to discuss possible options for independent candidates.
On Monday evening, Kristol noted
that Trump had been paying attention to his missives, tweeting:
"I'm traveling, so hadn't realized I'd so upset @realDonaldTrump. I'm sorry the mere mention of an independent candidate has so unnerved him."