It is thought that the transfer could now be completed quickly after the clubs finally agreed a fee following months of uncertainty.
Reports suggest that a compromise has been reached that will see Real pay an initial fee of £23.6 million, plus a further £3.9m in add-ons.
Tottenham are believed to have accepted Real’s offer now to give them enough time to sign a replacement before the window shuts at the end of the month.
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 move on.
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.
Wigan chairman Dave Whelan, meanwhile, has accused Chelsea of “messing about” over their pursuit of Victor Moses, warning them to pay the asking price or forget signing the forward.
The Latics are said to be holding out for a fee in the region of £9m for the 21-year-old forward Moses, who is out of contract next summer.
Whelan has claimed Chelsea have made an offer which includes add-ons based on appearances for the former England U21 international, who scored six Premier League goals last season.
He said: “Chelsea have got to make their mind up — yes or no — and at present they’re messing about a little bit which is unusual for Chelsea because they’re up front if they want someone. They would usually go and make a bid and get the player. With Victor, Chelsea have come in with offers including payments later, etc. It’s unusual for Chelsea because they’re usually a very straightforward club.”