The Argentine was expected to leave the Bernabeu last summer following the arrival of Gareth Bale, but was kept at the club following Mesut Ozil’s £42.5m switch to the Gunners.
Di Maria has since proven his worth under Carlo Ancelotti, helping Real to book a spot in the Champions League final.
But he is still seen as an understudy to Bale and Cristiano Ronaldo, and could be sold in order to fund a massive move for either of Liverpool’s Luis Suarez or Sergio Aguero of Manchester City.
This has alerted Arsenal and City, who are both thought to be weighing up £32m offers.
As well as the Premier League duo, Monaco and Paris Saint-Germain are also keen and have the financial clout to lure the South American to France.
The 26-year-old himself may well be open to a switch, with regular playing time for his peak years thought to be a priority.
Di Maria moved to Real in 2010 from Benfica, but has failed to live up to his billing as one of European football’s most exciting attacking players in the Spanish capital.