President Donald Trump’s personal attorney Michael Cohen helped a law and lobbying firm land a corporate client with ties to Kushner Cos., the family company of White House adviser Jared Kushner that is currently the subject of a federal probe.

Under a 2017 contract with Squire Patton Boggs, Mr. Cohen was paid $500,000 a year to help it drum up business, prosecutors said in a recent filing in federal court in New York, where Mr. Cohen is under investigation for potential bank fraud and campaign-finance violations. He also received a cut of any fees it collected from clients he referred.

Among five clients Mr. Cohen delivered to Squire Patton Boggs—before the firm terminated the contract with him in early March—was U.S. Immigration Fund LLC, according to court filings and Nicholas Mastroianni II, U.S. Immigration Fund’s chief executive. The Florida company connects businesses with foreign investors through a U.S. visa program.

After Mr. Trump’s election, Mr. Cohen began generating business both through his relationship with Squire Patton Boggs and through his own firm, Essential Consultants LLC.

Mr. Cohen formed Essential Consultants in October 2016 and used it to pay $130,000 to Stephanie Clifford, an adult-film actress known professionally as Stormy Daniels, to prevent her from publicly discussing an alleged sexual relationship with the president. Mr. Trump denies that relationship.