Kessie, 20, was widely reportedly to be having a medical ahead of joining Milan yesterday.
But, according to TMW, the tests picked up a "slight fitness defect".
The midfielder will reportedly undergo further examination later today.
All being well, TMW claim Kessie will complete a move to the San Siro on a five-year deal worth over £1.5m-a-year.
Kessie enjoyed a breakthrough campaign in Italy this season.
The Ivory Coast international made 31 appearances in all competitions for Atalanta, scoring seven goals and assisting four in the process.
His future has consequently been the subject of much speculation with both Chelsea and Manchester United credited with interest in taking him to the Premier League.
Kessie previously talked up a move to England, saying: "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."
However it appears that the Premier League duo have been beaten to the punch by Milan.
The Italian giants were purchased by wealthy Chinese owners in April.
And La Gazzetta dello Sport claim they fully intend to make their presence felt in the transfer market this summer by spending more than £100m on top talent.
