The deal for the versatile attacker is worth a reported €25 million (£22m/$29m).
The 25-year-old has agreed terms on a five-year deal to join up with new coach Carlo Ancelotti, who was appointed in place of Maurizio Sarri last month.
Verdi came through Milan's youth system but made just two appearances for the club before moving on to Torino.
Having also played for Carpi and Empoli, as well as a brief stint in La Liga with Eibar, the forward has spent the past two seasons with Bologna, scoring 10 goals in 34 Serie A appearances in 2017-18.
Verdi has won four caps for his country, the last of them coming in the friendly draw with Netherlands on June 6.