WASHINGTON—President-elect Donald Trump said he doesn’t “feel very strongly” about further investigations of onetime rival Hillary Clinton, backing away from fiery campaign rhetoric in which he pledged to appoint a prosecutor to probe her email server and foundation work.

“I don’t want to hurt the Clintons, I really don’t,” Mr. Trump said in an interview with the New York Times on Tuesday. When pushed to say whether he had ruled out prosecution of Mrs. Clinton, he said: “It’s just not something that I feel very strongly about,”...