Bale has consistently been linked with a move to La Liga during the transfer window, although Spurs manager Andre Villas-Boas has denied claims a deal is done.
The contract is yet to be signed but the transfer has moved a step closer to being finalised, Sky sources understand.
A world record breaking fee of as much as £94million has been suggested for the Wales international and the money from the sale would help finance a busy summer of transfers for Villas-Boas' side.
Real Madrid briefly put No 11 shirts bearing the name of Bale up for sale in their online store before the gaff was spotted and the website brought down.
Tottenham have spent £60million to bring four new signings to White Hart Lane, including a club-record £26million on Valencia striker Roberto Soldado.
But that figure could be smashed again, with a fee in the region of £30million likely to be required to attract Anzhi Makachkala's Brazilian midfielder Willian to north London.
Spurs have also been linked with a move for Roma and Argentina forward Erik Lamela.
Bale will play no part in the first leg of Tottenham's Europa League play-off against Dinamo Tbilisi on Thursday because of a foot injury.
Speaking on Wednesday ahead of the game in Georgia, Villas-Boas dismissed reports that Bale was closing in on a move to Real.
"That is not true," said the Tottenham manager.
While Bale has been recovering, Tottenham have spent £60million on four new signings and they could shell out that same figure over the next two days to land Willian and Lamela.
There are two schools of thought when it comes to the reasoning behind the record splurge by Daniel Levy. One is that he wants to bring the stellar signings in to persuade Bale to stay and the other is that the chairman is already spending the world-record fee the Londoners would receive if the Welshman left for the Bernabeu.
Villas-Boas was asked which one was the case and was reluctant to say that the new signings had been bought in order to try to tempt Bale to stay.
"Ideally we want to keep everyone to make us stronger - that is the ultimate objective of the club," the Spurs manager said.
"In the end, anything can happen. We are dragging ourselves into the last weeks of transfer activity and it can become very, very frantic."
Spurs expect to complete the signing of Willian after the Brazilian passed his medical while there is excitement among supporters about the prospect of Argentinian midfielder Lamela and Real Madrid left-back Fabio Coentrao also joining the club.