The experienced midfielder is on the verge of joining the Rossoneri after passing his medical at the weekend.
The Ghana international travelled to Italy on Friday evening after Chelsea sanctioned his departure to the San Siro side.
The first part of his medical exposed a knee problem that initially seemed to pose a problem, but Essien eventually got the green light after additional tests in the afternoon.
"Haters will broadcast your failure but whisper your success. Medical's done, looking forward to start my new life with AC Milan," the midfielder stated on his official Twitter account.
Essien becomes Milan's third signing of the January transfer window following the arrivals of Adil Rami and Keisuke Honda.
The 31-year-old started his professional career with Bastia and also wore the Olympique Lyonnais jersey before joining Chelsea in 2005.
He won two Premier League titles, four FA Cups and the Champions League during his time with the Blues.
The midfielder was shipped out on loan to Real Madrid last season and will now continue his career in Serie A.