Paris Saint-Germain and Chelsea are also attempting to sign the 26-year-old Croatia international but it is understood that a deal could be struck with Real in the next few days.
Real had cooled their interest, after an inquiry earlier this summer, in which they had offered cash plus Turkey midfielder Nuri Sahin but talks are understood to have resumed over a straight cash offer.
Spurs chairman Daniel Levy has been quoting an asking price of £50 million for Modric but the Real offer is expected to be the best one he will receive with the player preferring to move to Spain rather than join PSG.
Levy is adamant that if Modric quits Spurs then he will also leave the Premier League and the chairman will not even countenance an offer from Chelsea whose interest may, indeed, wane now that they have agreed a deal to sign the exciting young Brazil playmaker Oscar for around £22 million.
That move came about after Chelsea hijacked Spurs’ bid to sign the 20 year-old from Internacional and should Modric go it is likely that Spurs will make a move for Joao Moutinho although they will not meet Porto’s asking price of more than £30 million for the Portuguese midfielder. A deal, however, does seem likely for Moutinho to eventually move to White Hart Lane.
On Wednesday Spurs’ new head coach, Andre Villas-Boas, confirmed that the club would listen to offers for Modric after he once again informed them he would like to leave. “The interest Luka is getting from a number of different clubs is the same as he got last year,” Villas-Boas said.
“Luka is a player of major importance for the football club and that is why he attracts so many clubs. At the moment, it’s in the hands of the chairman to accept the right proposition.
"Chelsea was involved last year so there is nothing new in the situation. If the player’s ambition is to move to another club, we can respect that decision, but there are club interests that the chairman will defend and I think only when the right offer is met, there will be more news.”