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 years of loyalty to the former real-estate developer, whom he served for nearly a decade at the Trump Organization.

The White House didn’t respond to a request for comment, and there has been no indication Mr. Trump is planning to pay for his former longtime lawyer’s legal fees.

The Trump campaign had been footing the bill for some of Mr. Cohen’s legal expenses, paying nearly $230,000 to McDermott, Will & Emery LLP between October 2017 and January 2018, according to Federal Election Commission records and a person familiar with the matter. But those payments helped cover Mr. Cohen’s legal representation in the separate Russia investigation, not in the probe of his business dealings.

The Trump family had also financed a portion of Mr. Cohen’s legal bills in the latter probe, paying for the review of documents seized by prosecutors in an April raid of his properties, but hadn’t agreed or offered to cover his full fees, according to people familiar with the matter.