Mr. Cohen’s tough-guy routine raised some eyebrows. In 2012, he horrified a campaign aide for Mitt Romney, pulling up a pant leg to show off his pistol during a fundraiser before stashing the gun in a car. Another time, he tried to barge past Mr. Trump’s security chief, Keith Schiller, after being informed that the boss was in a meeting. Mr. Schiller threw him to the ground and said, I told you, you’re not going in.

Mr. Trump kept Mr. Cohen around but didn’t seem to respect him. When Jerry Falwell Jr., son of the evangelical leader, told Mr. Trump during the 2016 presidential campaign that he and his wife really liked Mr. Cohen, Mr. Trump responded, Really? Really?

Mr. Cohen played a behind-the-scenes role in Mr. Trump’s presidential bid, to the chagrin of the actual campaign staff. For the launch event, Mr. Cohen proposed bringing an elephant to Trump Tower on Fifth Avenue. After that was rejected, he coordinated with a friend—whom Mr. Trump had nicknamed “The Screamer” for his boisterous voice—to pay 55 actors in cash to attend the event.