The fee has yet to be confirmed but both clubs have confirmed the deal and he will be presented to Real's supporters at 12pm on Monday.
However reports in Britain say that the price paid is actually €100m (£85.3m) which would be a record. There are some indications that this figure will not be announced publically as Real Madrid would rather Ronaldo remains the official world record signing.
“I am not sure there is ever a good time to leave a club where I felt settled and was playing the best football of my career to date. I know many players talk of their desire to join the club of their boyhood dreams, but I can honestly say, this is my dream come true," said Bale.
"I am well aware that I would not be at the level I am today were it not for firstly Southampton and then Spurs standing by me during some of the tougher times and affording me the environment and support they have.
“Tottenham will always be in my heart and I’m sure that this season will be a successful one for them. I am now looking forward to the next exciting chapter in my life, playing football for Real Madrid."
Tottenham chairman Daniel Levy added: "'Gareth was a player we had absolutely no intention of selling as we look to build for the future.
"He is a player whose career we have fostered and developed and he was only a year into his new four year contract.
"Such has been the attention from Real Madrid and so great is Gareth's desire to join them, that we have taken the view that the player will not be sufficiently committed to our campaign in the current season.
"'We have, therefore, with great reluctance, agreed to this sale and do so in the knowledge that we have an exceptionally strong squad."
Bale will wear the number 11 shirt in Madrid.
Bale joined from Southampton in May 2007 and during his six years with the club made 203 appearances, scoring 55 goals.
Speaking after Spurs' 1-0 defeat to Arsenal in the North London derby, Tottenham manager Andre Villas-Boas said: "He is going to join Real Madrid and we wish him all the best.
"He has left us with great memories from last season which I think every Tottenham fan cherishes but he has moved on and we move on as well."