According to reports in the Italian media, the 31-year-old Ghana international failed tests related to an old knee injury.
But Essien, who arrived in Milan on Friday evening, has now said that the medical was completed as a success.
“Haters will broadcast your failure but whisper your success. Medicals done, looking forward to start my new life with AC Milan,” Essien posted on his Twitter page.
Essien underwent stress tests at the Milan Lab on Sunday in a deal that is free for the Italian club.
Earlier, Antonio Nocerino moved to West Ham on loan in order to make way for Essien.