If the move goes through it would bring to an end a four-year spell at White Hart Lane for Modric, during which time he has developed into one of the Premier League's most impressive performers.
The Croatia international was linked with a move to Chelsea last summer but Spurs chairman Daniel Levy stood firm to keep him at the club.
However, intense speculation resurfaced ahead of the current transfer window and Modric made no secret of his desire to join Real Madrid, with Sky Sports now claiming a deal could be completed quickly.
New Tottenham manager Andre Villas-Boas criticised Modric after he did not travel to the United States for their pre-season tour, and the 26-year-old has continued to train apart from the rest of the squad.
Villas-Boas has already strengthened his midfield with the signing of Gylfi Sigurdsson since replacing Harry Redknapp at the Tottenham helm, while Joao Moutinho has been linked with a move to join up with his former Porto manager.
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