Michael Cohen has hired New York lawyer Guy Petrillo to represent him in a federal investigation into his business dealings, and has told associates he wants President Donald Trump, his former boss, to pay his legal fees, according to people familiar with the matter.

Mr. Cohen has frequently told associates in recent months he is frustrated that the president hasn’t offered to pay his legal fees, which he has said are “bankrupting” him, according to one of the people. He has said he feels that Mr. Trump owes him after his...