WASHINGTON—Deputy Attorney General Rod Rosenstein pressed White House counsel Don McGahn to correct what he felt was an inaccurate White House depiction of the events surrounding FBI Director James Comey’s firing, according to a person familiar with the conversation.

Mr. Rosenstein left the impression that he couldn’t work in an environment where facts weren’t accurately reported, the person said. The deputy attorney general objected to statements by White House aides citing Mr. Rosenstein’s critical assessment of Mr.Comey’s...