Mr. Cohen was supposed to pay Mr. Gauger $50,000 for the work, the man told The Journal, but the full amount was never paid out. However, Mr. Cohen billed the Trump Organization $50,000 for technology services, according to federal prosecutors in Manhattan, who referenced the amount in a charging document.

Mr. Cohen did not respond to a message seeking comment. Mr. Gauger could not be reached for comment.

According to The Journal, Mr. Cohen also asked during the 2016 campaign for Mr. Gauger’s help establishing a “Women for Cohen” Twitter feed that described Mr. Cohen as a “sex symbol,” and sought to promote his public appearances. Mr. Cohen was a frequent cable news presence during the 2016 campaign in support of Mr. Trump.

Mr. Cohen has agreed to testify before the House Committee on Oversight and Reform on Feb. 7. Mr. Cohen’s testimony will be restricted to avoid issues directly related to the special counsel investigation into whether there was collusion by anyone in the Trump campaign with Russian officials.

Over the weekend, Mr. Trump, in an interview on Fox News, tried to urge prosecutors and members of the media to investigate Mr. Cohen’s family, and he blasted his former fixer as “weak.”