The 31-year-old arrived in Italy on Friday to undergo a medical and finalize personal terms with the Rossoneri after Chelsea boss Jose Mourinho sanctioned the player's departure.
Essien himself said he was relishing the prospect of working under new head coach Seedorf, and Galliani believes the Ghanaian could prove a pivotal player in the Dutchman's system.
"The pursuit [of Essien] lasted a few months," he told reporters in Milan. "I think Essien can do well in Clarence's new model. He is perfect to play in front of the defense, and he can play in the Champions League.
"From a physical point of view, I think Essien is reliable. He must pass the medical tests at the Milanello, then there will be a signing."
Galliani went on to reveal that Sulley Muntari proved instrumental in encouraging Essien to make the move, and hinted that Milan could see further activity in the final week of the January transfer window.
"Muntari was one of his greatest sponsors. Riccardo Montolivo will be out for two Champions League games and Muntari one, so we needed a midfielder," he added.
"More activity in the market? Maybe... we'll see."