The Atalanta midfielder enjoyed an impressive season scoring seven goals and providing three assists.
And it was that form which led to him being linked with a move to the Premier League, in particular Chelsea and Manchester United.
Earlier in the year, Kessie spoke of being flattered by their interest.
He said: "I am very flattered that clubs of this level are interested in me. This proves that I’m having a good season, that I continue to progress. I don’t want to go too fast.
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Chelsea? That is also a big club, nobody would turn them down, but I dream of United
"I like the Premier League and in particular Manchester United, the club I dream of playing for.
"Chelsea? That is also a big club, nobody would turn them down, but I dream of United."
But it now appears as though Kessie sees the next step in his career at the San Siro.
Franck Kessie is on his way to AC Milan
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Reports in Italy say Milan reached an agreement with Atalanta to sign the Ivory Coast international for around £24million and the player is having a medical this morning.
Kessie will then officially join when the transfer window opens in July.
Milan are also closing in on Argentine defender Mateo Musacchio from Villarreal and Wolfsburg full-back Ricardo Rodriguez.