The Ivory Coast international has been hugely impressive since his arrival at San Siro, scoring 31 goals from 188 appearances in all competitions.
However, the 24-year-old's contract is due to expire at the end of the current campaign, inevitably leading to widespread interest in his signature.
Despite both parties holding a desire to renew terms, Calciomercato claims that they have reached an impasse in negotiations.
The report adds that Milan and Kessie still remain calm about the situation with more than two months left before the player can officially communicate with other clubs.
Should no deal be agreed by January, Milan would have to consider parting ways with one of their key man in order to recoup a fee for his services.