The 24-year-old enjoyed a remarkable season with the north Londoners in 2012/13, bringing him to the attention of the Spanish giants.
Madrid are reportedly willing to better the world record fee of ?80million they shelled out on Cristiano Ronaldo four years ago, with Bale thought to be keen on a switch to the Bernabeu.
Tottenham did not wish to comment on the speculation when contacted by Press Association Sport and Real Madrid manager Carlo Ancelotti was also coy when asked about the Welshman.
"We will see what happens," the Italian said. "It is between the two clubs. Right now our team is very strong and we will have to see what happens in the future."
There are clearly hurdles to overcome but, should Bale make the move to Los Blancos, former Spurs boss Redknapp believes the winger would shine in the Spanish capital.
"He is behind Messi and Ronaldo in terms of being the best player in the world," said Redknapp, who coached Bale for four years. "He is at a good club but the opportunity to play for Real Madrid does not come along too often. He has probably dreamt about that since he was a little boy.
"It seems they have offered fantastic money so it could happen, but I don't know. I wouldn't be sure. If he goes there, he will be a great success there. I am sure of that.
Redknapp, now manager of SkyBet Championship promotion hopefuls QPR, helped transform Bale into the player he is today.
Originally a left-back with Southampton, Redknapp pushed him further forward and was at the helm when he won the Professional Footballers Association Player of the Year award for the first time in 2011, a feat he repeated last season.
Source: PA
Source: PA