"I'm happy to be here," he told Milan Channel. "I will do my best for Milan and I hope to be ready to play.
"I spoke with Mr Seedorf two days ago. I cannot wait to be available.
"I really want to thank the [Chelsea] chairman (Roman Abramovich) and the team and [manager Jose] Mourinho, who has allowed me to wear the Rossoneri shirt."
The completion of the transfer is subject to a medical and Essien agreeing personal terms with Milan.