Cardinale has not given up on bringing Krösche to the club, according to journalist Matteo Moretto, months after talks over the Eintracht Frankfurt sporting director first stalled.
Cardinale, whose RedBird fund controls the Rossoneri, had already identified Krösche as his preferred appointment over the summer, after an earlier attempt to hire Ralf Rangnick fell through.
The American, who wants a "world-class" sporting structure at AC Milan, is said to remain in regular contact with Krösche's camp, and the German's position at Eintracht has looked less secure since his relationship with that club's supporters cooled.
The pursuit of executive reinforcements comes as head coach Rúben Amorim works to fix the issues in his own squad. Despite criticism, the Portuguese is trying to correct the aspects of his system that are not working, particularly getting the ball to his forwards more effectively; Gonçalo Ramos has scored only once so far and has had few clear chances to add to that tally.
Amorim also needs to sort out the team's tactical balance, but confidence in his project is said to be growing, helped by a strong response from senior players such as Christian Pulisic in training and matches alike after a difficult defeat to Benfica.