Gov. John Kasich is no fan of Senator Ted Cruz, but he does not want his allies borrowing one of Donald J. Trump’s go-to insults.

Mr. Kasich on Friday called on a “super PAC” supporting him to take down an advertisement describing Mr. Cruz as “Lyin’ Ted,” a nickname for the Texas senator favored by Mr. Trump.

The commercial, released by New Day for America, depicts Mr. Cruz with a Pinocchio-like nose growing and wrapping around his neck.

Among other things, it refers to an ad by a super PAC supporting Mr. Cruz that inaccurately asserted that the liberal billionaire George Soros had provided hundreds of thousands of dollars in support of Mr. Kasich.

“If Ted Cruz’s mouth is moving, he’s lying,” the ad concludes.

On the campaign trail, Mr. Kasich boasts about how he avoids personal attacks. Speaking to reporters in Pennsylvania on Friday, he said he had expressed his displeasure with how Mr. Cruz was described in the ad. (Top Kasich aides used the same nickname, “Lyin’ Ted,” in Twitter posts this week, which Mr. Kasich also found objectionable.)

“I would hope that they will not use that word and they’ll take it down,” Mr. Kasich said of the commercial.

At the same time, Mr. Kasich displayed no affection for Mr. Cruz or his allies, bringing up a Wall Street Journal editorial that described the Soros ad as a “smear.”

“The problem with Senator Cruz is he has no record,” Mr. Kasich added. “His record is shutting down the government and making everybody he works with upset.”

Super PACs, which can accept unlimited donations, are not allowed to coordinate with candidates’ campaigns, and this is not the first time Mr. Kasich or his team has criticized an ad from the super PAC supporting his run. Before the New Hampshire primary, the Kasich campaign called on the super PAC to pull an ad that went after Senator Marco Rubio of Florida; the group obliged.

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Asked about Mr. Kasich’s comments, Connie Wehrkamp, a spokeswoman for the super PAC, responded by expressing displeasure about the Soros commercial.

“We stand by our ad and the fact that what Ted Cruz’s PAC ran was completely false,” she said. “We will always call out our opponents if and when they do that, but we are going to review the language we used.”