The Ghana international, who spent last season on loan at Real Madrid, is set to pen an 18-month contract at the San Siro.
If he joins the Serie A side he will become Clarence Seedorf's first signing as AC Milan manager.
Essien arrived in Milan on Friday evening and he told the Milan Channel: "I'm happy to be here. I will do my best for Milan and I hope to be ready to play.
"I spoke with Mr (Clarence) Seedorf (the Milan head coach) two days ago. I cannot wait to be available.
"I really want to thank the chairman and the team and thanks to Jose Mourinho, who has allowed me to wear the Rossoneri shirt."
Chelsea boss Mourinho was asked about Essien's future ahead Chelsea's FA Cup tie against Stoke.
He said: "Michael Essien is in a position at the moment where he can go or he can stay. If he stays we will be more than happy.
"He is a fantastic club man and a fantastic team player.
"But if something big comes for him he is another one who we feel we must show the respect, and to show him the respect is not to stop him from going."
AC Milan said in a statement on their website: "Just turned 31 and with a CV including domestic title and Champions League wins as well as victory with the Ghana national side, Michael Essien is expected in the city this evening."
Essien began his career in Europe at French side Bastia after moving from his native Ghana at the age of 17.
He moved to Lyon in 2003 where he established himself as one of the most sought-after defensive midfielders in world football.
He won the Ligue 1 Player of the Year award prior to signing for Chelsea for a then club-record fee of £24.4m in the summer of 2005.
Despite winning eight major trophies for Chelsea, Essien's career in England was blighted by serious knee injuries, although he has made over 250 appearances during his Stamford Bridge career.