In an e-mail exchange with Chicago Tribune sportswriter Rick Morrissey, Cuban said he believed he sent the application in last week, although he wasn't sure of the date.
The Tribune Company announced in April that it intended to sell the National League team.
Several possible bidders for the Cubs have recently emerged.
Morrissey reported that John Canning heads a private-equity firm based in Chicago and is considered a strong candidate.
Other potential candidates include Chicago Wolves owner Don Levin. The Omaha-based Ricketts family is also reportedly preparing an offer.
The Ricketts' bid will reportedly be led by Tom Ricketts, who studied at the University of Chicago.