The 36-year-old defender ended his time with the Nerazzurri after helping them win the Scudetto and is now considering the options available for the next stage of his career.
Bologna are among the clubs assessing whether to bring the experienced full-back to Emilia-Romagna. He is now training with the club.
According to journalist Nicolò Schira, Bologna have been offered the opportunity to sign Darmian, with the club's interest taking on added significance following Emil Holm's injury.
Darmian's ability to operate on the right side of defence could give Bologna another experienced option as they deal with Holm's absence.
The former Italy international has spent much of his career operating at full-back but can also provide versatility across the defensive line.
No agreement has been reached at this stage, but Bologna have the opportunity to bring a player with extensive Serie A and European experience into their squad.
Darmian's availability without a transfer fee is another factor for clubs considering a move, with his next destination yet to be decided.
Bologna must now determine whether to turn their interest into a concrete proposal as they look for a solution following Holm's injury.