The French-born Guinean midfielder has joined the Rossoneri outfit on a one-year loan deal from Genoa.
Milan have an option to sign Constant outright at the end of next season for an undisclosed fee.
"I am delighted to be at Milan," Constant said to Sky Italia after undergoing a medical.
"I knew about Milan's interest two weeks ago and now I am here.
"I can't wait to play and prove my value at Milan. I hope I can win a title with Milan next season."
Constant only moved to Genoa from Chievo last summer in a seven million euros transfer deal.
The 25-year-old is considered the ideal replacement for Sulley Muntari, who will be out for six months after undergoing knee surgery to repair a torn cruciate ligament.