Professional football and baseball players have long made money from their likenesses in videogames and on trading cards. Now they are teaming up with a private-equity firm in an unprecedented deal to amp up that revenue and invest it.

The players associations of the National Football League and Major League Baseball have struck a deal with RedBird Capital Partners to form a new company called OneTeam Partners, which will be centered around the management of the players’ portrayals, the parties said.