Hurriyet reports that Galatasaray have already reached an agreement with Real Madrid on the transfer fee.
The Spanish side have given Galatasaray permission to talk to Di Maria over a possible move.
Arsenal, Manchester United and Tottenham have also been strongly linked with a move for Di Maria.
Real Madrid could be forced to cash in on Di Maria in order to raise some of the expected £86million they are expected to pay for Tottenham winger Gareth Bale.
