According to Pete Jensen of The Telegraph, the Spanish club has asked Spurs to contact them if Bale should become available and to have the opportunity to match any offers from other clubs.
Further, Madrid president Florentino Perez is apparently considering a summer bid for the winger that is somewhere in the range of £50 million to £60 million.
Bale has become one of the hottest commodities in the world over the past few weeks, scoring seven beautiful goals in his last five matches and carrying the load for both Tottenham and Wales.
With the top clubs surely set to make bids for the Welshman this summer, Real Madrid are looking to exploit their "special partnership" with Spurs, established in last summer's Luka Modric deal, to gain a leg up on their rivals.
While it is unclear how Madrid would be able to make such a huge deal, given their debt and the coming UEFA crackdown on financial fair play, it is quite understandable for Tottenham to be tempted by the money offered.
Is there a price at which you would want to see Tottenham sell Bale?
After all, chairman Daniel Levy has always been a very down-to-earth administrator, unafraid to sell his best players if offered a sum that he believes to exceed their value.
While Gareth is clearly one of the finest players in the world, a fee that would rival many of the highest fees in history (before negotiation) could mark an opportunity to reinvest in the side and create an even better team.
No one wants to see Gareth Bale leave White Hart Lane, but don't be surprised if the Welshman takes his talents elsewhere should other clubs start bidding in excess of £50 million.
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