According to French newspaper L'Equipe and also reported by A Bola, Monaco are strong contenders to land the Argentine international this summer - but they face competition from United, Arsenal and Manchester City among others.
Juventus and Paris Saint Germain are also tracking developments at the Bernabeu, with Di Maria believed to be available this summer.
He enjoyed an excellent season under Carlo Ancelotti last term, scoring four goals and setting up 17 more in 27 starts for Real Madrid. He also played an important role as the club won it's tenth Champions League trophy.
But Ancelotti is reportedly willing to part with his services if Real Madrid receive a substantial offer for the winger.
A bid around the €50m mark could do it, although it remains to be seen whether any of the suitors would be willing to go that high.
United, Arsenal and City certainly have money to spend this summer, but a move for Di Maria would take a significant chunk of their transfer budgets and both Arsenal and United have more pressing concerns elsewhere.
Meanwhile Paris Saint Germain don't look like pausing their spend even in the wake of Financial Fair Play sanctions. They look set to complete a deal for David Luiz and have been linked with a number of other expensive additions.