It represents a shift in stance after they had previously privately indicated they were not interested in the player.
Argentina international Di Maria, 26, recently told the La Liga club he wants to leave after rejecting a new deal.
A switch to Louis van Gaal's side would see the player follow in the footsteps of international team-mate Marcos Rojo, who signed for United on Wednesday.
During the summer United officials distanced themselves from a move for Di Maria, who missed the World Cup final through injury.
Madrid paid £20m to bring him to the Bernabeu from Benfica in 2010.
United, who play Sunderland on Sunday, lost their opening Premier League game 2-1 at home to Swansea last weekend.
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