The clubs agreed a deal to take the Ghana midfielder to Italy on Friday, although rumours of an unsuccessful examination due to long-standing knee issues surfaced soon afterwards.
Essien has struggled for first-team action in recent years at Stamford Bridge, having made just 19 Premier League appearances since the 2011-12 season, and spent last term on loan at Real Madrid under now Chelsea manager Jose Mourinho.
And his Milan move looks set to go ahead after he claimed to have passed his medical at the Serie A club.
He posted: "Haters will broadcast your failure but whisper your success.Medicals done,looking forward to start my new life with AC Milan.#jahbless."
Essien's first league start of this season hinted at his decline as an error helped Southampton take a 1-0 lead in the opening minute of December's clash - although Chelsea recovered to register a 3-1 home win.
Having made just five Premier League outings this term, Nemanja Matic's return to Stamford Bridge pushed the midfielder further down the pecking order, and Essien has thanked Mourinho for the opportunity to resurrect his career at San Siro.
