Michael Avenatti tried to make a deal last week in one of the few businesses that can be dirtier than national politics: college sports.

Mr. Avenatti rose to prominence representing the former adult-film star Stormy Daniels regarding her alleged affair with President Donald Trump. But last week, a day after he allegedly threatened Nike Inc.’s outside counsel that he would “take 10 billion dollars off your client’s market cap,” he walked into the New York offices of the celebrity attorney Mark Geragos to resolve another case...