It had been reported that the first part of his examination with the Italian club's medical team had revealed a knee problem, which some had speculated could have thrown the deal into doubt.
However, the 31-year-old took to Twitter this afternoon to dispel the rumours.
"Haters will broadcast your failure but whisper your success. Medicals done, looking forward to start my new life with AC Milan," wrote the Ghana international.
It is believed that Essien will sign an 18-month deal at the San Siro.