“Yawn... Welcome to Minneapolis where we pay our bills, we govern with integrity, and we love all of our neighbors,” Mr. Frey wrote, seemingly referring to reports that the Trump campaign has yet to pay some cities for such security fees.

As of June, at least 10 cities were still waiting on security payments for Trump campaign events, according to the Center for Public Integrity, a nonprofit news organization based in Washington. The largest outstanding payment was for more than $470,000, which El Paso said it was still owed for a February rally. The oldest outstanding payment was for an event held in Burlington, Vt., in January 2016. (That bill had not yet been paid as of Tuesday, a city spokeswoman said.)

Mr. Trump’s tweets on Tuesday morning were in response to a Fox News interview with Lt. Bob Kroll, the president of the Minneapolis Police Federation. The union is selling “Cops for Trump” T-shirts in response to a reported Police Department policy barring officers from wearing uniforms in political ads or at political events in support of a candidate. In response to the segment, Mr. Trump thanked the union; shared a link to its website, where the shirts are sold; and posted a video of the interview.

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In its news release, shared online on Monday night by Mr. Trump’s campaign manager, Brad Parscale, the campaign described the security fee for Thursday’s event as “ridiculous.”

In an attached letter to AEG Worldwide, the campaign’s lawyers disputed the company’s apparent contention that the campaign must cover security costs for the event.