The 25-year-old's existing deal at the San Siro expires at the end of the season and he has so far resisted all attempts to agree an extension.
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One of the main sticking points has been Kessie's proposed salary with Milan's latest offer at €6.5million-a-year, according to ESPN sources.
Sources have told ESPN that Spurs are preparing to indicate they will surpass that figure with chairman Daniel Levy ready to sanction the deal given there would be no transfer fee involved.
Kessie will be free to negotiate directly with overseas clubs from Jan. 1 and sources told ESPN there are multiple clubs closely monitoring the situation.
Chelsea, Liverpool and Manchester United are expected to strengthen their midfield options either from January or in the summer but as yet there has been no firm indication the Stamford Bridge have formally entered the running. Milan have also not given hope that Kessie will commit his future to the club.
Sources have told ESPN that Conte will be given significant funds to overhaul the squad but the bulk of that work is expected to take place in the summer given how difficult business can be to complete in January.
Speaking on Thursday, Conte confirmed he has completed his evaluation of the squad after succeeding Nuno Espirito Santo as head coach in November and cited a need to strengthen in January.
"After only two weeks, I said we need to improve the quality of the squad," he said. "For sure we need to improve the quality of the squad.
"I was sure after seven days, 10 days, two weeks about this. And then I make the evaluation who are the players I can count on and the players I can't count [on]."