The Rossoneri are on the lookout for a new central midfielder before the closure of the transfer window and Bakayoko has emerged as a target for coach Gennaro Gattuso.
The 23-year-old has struggled to make an impact at Chelsea after joining them from Monaco last summer for a fee of 45million euros, and the Blues are ready to let him move on.
Chelsea are now coached by former Napoli boss Maurizio Sarri and it is thought that Bakayoko does not fit in with his style of play and plans for the future.
Milan have been priced out of a move for Sergej Milinkovic-Savic and Bakayoko appears to be a much more affordable option for the San Siro outfit.
Chelsea would prefer to sell but are also happy to loan out the tough-tackling midfielder with a view to a permanent deal.