The same source has suggested that the player has already agreed a long-term deal with the French giants, with los Blancos also set to accept a €35 million offer from the Principality Club.
The January transfer, therefore, looks on the cards, and Real Madrid boss Carlo Ancelotti is the only one who could still pull the plug on the deal.
He reportedly values the Argentine highly and could ask the club chiefs not to sanction his sale to Monaco.
Angel Di Maria has been struggling for playing time since the arrival of Gareth Bale and he is hoping to leave the club in January in search of first team football.
Having closed the gap behind leaders Paris Saint-Germain to just two points, Monaco are 5/1 to win the league title.