Turkish website Hurriyet are reporting that Galatasaray have agreed a deal with Madrid for the in-demand midfielder, though the player remains unconvinced that a move to the Türk Telekom Arena is in his best interests.
The 25-year-old isn't in newly-appointed Real Madrid boss Carlo Ancelotti's first-team plans, and has been widely tipped to secure a Bernabeu exit before the transfer window slams shut.
Manchester City and Manchester United had previously been linked with the Argentine international earlier in the summer, but it is Arsenal who remain keen and are the front-runners in the race for the player's signature.
Di Maria is not the only Real Madrid player on Arsene Wenger's transfer wish-list either, with the Arsenal boss also reportedly interested in Mesut Ozil and Karim Benzema, though the latter has since signalled his intentions to remain with the La Liga giants.
If Arsenal are to secure Di Maria this window, they must move fast to ensure they do not miss out on another transfer target, though a move is likely to be dependent on Gareth Bale completing his world-record move to Madrid.