Massimiliano Allegri's men have endured a dreadful start to the season and sit 19 points behind league leaders Roma in 10th.
Milan have not tasted victory in any competition since their 1-0 home win over Udinese on October 19.
Genoa's upturn in fortunes has coincided with head coach Gian Piero Gasperini's return to the Stadio Luigi Ferraris.
The visitors have lost just once since Gasperini's appointment on September 29, and sit seventh in Serie A as a result.
Gasperini's side have not conceded a goal in three games and will be hopeful of maintaining their strong defensive record against a Milan side that has struggled to find the net in recent times.
"We're expressing ourselves well at Genoa and playing good football," he told the club's official website.
"Gasperini and the results have brought enthusiasm, trust and determination. We want to defend our seventh-place position and give satisfaction to the fans."
