Kessie, 20, is set to become the first Rossoneri signing under the new Chinese owners and will cost the San Siro club €28 million.
The Atalanta midfielder is expected to sign a five-year contract worth €2.5m per year with Vincenzo Montella's club.
As well as attracting attention in Serie A, Kessie, who is an Ivory Coast international, had been linked with a move to Premier League giants Man United and Chelsea.
In February, the midfielder admitted that he dreamt of one day playing for the Red Devils and last month his agent George Atangana named Manchester City, Liverpool and Arsenal as potential destinations.
Roma, though, were identified as the suitors who had "the advantage" in the fight for the midfielder's services.
“Franck is close to many clubs and close to none," Atangana told Rai Sport in April.
"I must admit that Roma have the advantage for Kessie, but we haven’t closed a deal yet because there are other top clubs interested who continue calling me."
However, Milan have managed to beat their rivals in the race for the coveted starlet.
Kessie has starred for Atalanta in his breakthrough season, scoring six goals in 28 league appearances to propel his team up to fifth place in Serie A.