President Trump said that Defense Secretary Jim Mattis could be considering whether to step down from his post and that he sees the Pentagon chief as “sort of a Democrat,” a veiled critique of one of his most popular cabinet secretaries.

“I have a very good relationship with him. I had lunch with him two days ago,” Mr. Trump told “60 Minutes” in an interview being aired Sunday. “It could be that he is” leaving, Mr. Trump said, responding to a question about whether he wants Mr. Mattis to leave.

“I...