“It is, in fact, a campaign slogan, that is a campaign item, and it’s completely inappropriate for the troops to do this,” Mr. Kirby said. “Not supposed to do this. And I’m sure their boss is seeing that. They’re not going to be happy about it.”

CNN Pentagon correspondent Barbara Starr called it “very much against military policy and regulation.”

“The pool reporters traveling said that the troops brought the hats with them, including one hat that said ‘Trump 2020,’” said Ms. Starr. “We will have to see if that actually proves to be the case. The question is, if they brought them or if the President brought them: What commander allowed that to really happen?”