Talksport says that the bid of the city of £ 46 million has reached Milan's appreciation for the Dutchman.
Reijnders, 26, has already agreed personal conditions with City and is now due to a medical one.
The Holland International was named Serie A -Midfielder of the Year last season. Milan was resigned to losing Reijnders this summer after a gloomy campaign in the past in which the club missed the European competition.
The last term, Reijnders scored five goals and made five assists of 54 games in all competitions.