The Argentine winger faces being shipped out to pay for the former Atletico striker as Real seeks competition for Karim Benzema this season
Di Maria is a target for Paris Saint-Germain, but the French champion is yet to meet Madrid's valuation for the 26-year-old and the Argentine could yet stay at the Santiago Bernabeu.
However, Madrid is keen to push through a sale to recoup some funds in another intense period of summer spending, which has already seen the club pay €30m for Toni Kroos and €80m for James Rodriguez.
James' Colombia and former Monaco teammate Falcao could also arrive this summer for a fee in the region of €40m, but Madrid will wait until late in the window to see how he recovers from the anterior cruciate ligament rupture which saw him miss the World Cup.
"Right up until the last day something can happen," Madrid president Florentino Perez said this week. "Be it buying players or selling them."
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Last year, Real brought in Gareth Bale and sold Mesut Ozil right at the end of the window and this time the last-minute action could involve Di Maria and Falcao, with Levante goalkeeper Keylor Navas set to seal a move next week.
Di Maria is unhappy at being used as a transfer pawn for the second summer in a row, especially after his brilliant performances helped the club claim the Copa del Rey and the Champions League last term, and is set for showdown talks with the board when he returns to training on Aug. 5.
Madrid plays in six competitions next season and coach Carlo Ancelotti would prefer to keep Di Maria, but the Italian also knows Madrid needs another center forward after selling Alvaro Morata to Juventus earlier this summer.
Benzema is now the club's only senior striker and if the club decides to keep Di Maria (and sell Sami Khedira and Isco instead), Madrid may take Falcao on loan for a fee of €12m with a view to a permanent deal for around €35m next summer, allowing Ancelotti to assess his fitness after the injury and Perez to raise funds for the move in 2015.