The Rossoneri have a ‘concrete interest’ in 27-year-old Aurier and have already started talks.
Spurs value Aurier, who has two years left on his contract, at between €12m and €15m.
After losing prominence under Mauricio Pochettino, Aurier has been used more often by Jose Mourinho at Spurs.
Aurier has made 42 appearances in all competitions this season with a return of seven assists and two goals.
Even so, his future in London is uncertain and Milan are making a serious attempt to sign him.
Head coach Stefano Pioli wants to upgrade the right-hand side, where his current options are Davide Calabria and Andrea Conti.