The Bernabeu club see the Welsh winger as their next big star and are willing to break the bank to persuade him to leave the Premier League club.
It’s believed that Real and Spurs have a special partnership, that was agreed during Luka Modric’s summer move to Spain, which could help any potential deal.
The 23-year-old is one of the White Hart Lane club’s star names, but has previously said he could seek a new challenge in the future, hinting that he is happy to try his hand abroad.
With Cristiano Ronaldo’s immediate future far from certain, the Galacticos are keen to draft in a potential replacement.
Bale fits the bill, with his direct attacking approach making him a threat to even the sternest of opposition defences.
The ex-Southampton man has been in fine form so far this season, notching six goals and weighing in with a number of important assists.
Real Madrid are looking to raise the £56m kitty by selling Brazilian star Kaka and former Benfica winger Angel Di Maria.
Bale has been previously linked with Manchester United, but its thought that Tottenham would fiercely oppose any move to a domestic rival.