The Rossoneri have decided to promote the Primavera coach to the full team next season after deciding a move for Unai Emery would prove too costly.
The former Netherlands international took over at San Siro in January following the sacking of Massimiliano Allegri and delivered the club to an eighth-placed finish in Serie A.
However, he has not done enough to keep his job after the relationship between coach and board deteriorated during his five-month spell.
Milan has been searching for a replacement but has been put off its pusuit of Sevilla coach Unai Emery thanks to the high costs involved.
Club management figures, including owner Silvio Berlusconi and CEO Adriano Galliani, met Monday afternoon and have decided that Inzaghi is the only viable candidate to train the team next season.
The 40-year-old enjoyed a glittering 11-year spell at the Giuseppe Meazza, winning Serie A and the Champions League twice a piece.
Since his retirement in 2012, he has been in charge of the club's Primavera side and has long been seen as a future coach for the full team.
Berlusconi and Galliani are expected to inform the former Italy international of their decision to promote him imminently.