The 31-year-old has found opportunities limited at Stamford Bridge since returning from a loan spell at Real Madrid last season and has permission from Jose Mourinho to leave.
Milan were one of a number of interested parties and on Friday announced Essien is expected in northern Italy later in the evening.
A statement on acmilan.com read: "Michael Essien in expected in the city this evening."
Essien joined Chelsea from Lyon for £24.4million in 2005.
The midfielder suffered from a succession of knee injuries and the signing of Nemanja Matic from Benfica saw Essien slip further down the pecking order at Stamford Bridge.