The French-born Guinean player joins the San Siro outfit with Milan having an option to make his move permanent next summer.
The Serie A giants made this move in the transfer market after confirming that midfielder Sulley Muntari would be sidelined for six months through injury.
The Ghana international recently underwent an operation to reconstruct the anterior cruciate ligament in his left knee.
"It has been a quick decision to take Constant and I must thank (Genoa) president (Enrico) Preziosi," Milan vice-president Adriano Galliani told Gazzetta dello Sport.
"He was a player that we wanted to sign last year.
"In agreement with our coach, we have decided to get him once we found out about Muntari's injury.
"Muntari has had an ugly injury. I have never had a player that tore his cruciate (ligaments) during the holidays."
The Rossoneri club also confirmed they have acquired 50% of the sporting rights of defender Francesco Acerbi from Genoa.
Acerbi, who played at Chievo last season, has put pen to paper on a five-year contract with Milan.
"To play at Milan is a dream come true for anyone," said the 23-year-old Acerbi. "I will try to earn a regular spot in the team.
"Playing alongside Thiago Silva will no doubt help me as he is the best defender in the world."
Acerbi and Constant will undergo a medical this week before being unveiled to the media.