According to Tuttosport (via MD), Florentino Perez is calling for the club to sign the 26-year-old while looking for a long-term successor to Luca Modric.
Reijnders has been a highlight in a wrestling AC Milan side, scored 14 goals and the provision of three assists in his 46 games in all competitions.
Madrid is supposed to weigh their option in the middle of the park and to be linked to the real Sociedad star Martin Zubimendi.
AC Milan can be forced to cash in one of their valued assets because they have to miss European football next season.