The Welshman has agreed a six-year deal believed to be worth €10 million net a year and will be presented to the media and the club's fans after undergoing a medical in the Spanish capital.
Widespread reports in England have suggested that the move will make Bale the world's most expensive player with Real set to pay €101 million ($133.5 million, £86 million) for his services.
However, media in Spain insisted that Real will only pay €91 million, three million less than Los Blancos paid for Cristiano Ronaldo back in 2009.
"I would like to thank everyone at the Club, the Chairman, Board, staff, coaches and players - and most of all the fantastic fans who I hope will understand this amazing career opportunity," Bale said in a statement on the Spurs website.
"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.
"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."
Bale was coveted by the Spanish giants following a series of amazing performances for Tottenham last season that landed him both the Premier League players and football writers' player of the year.
He scored 21 goals in 33 Premier League appearances, but his efforts still weren't enough to lift Spurs into the Champions League as they were edged out of fourth place by just one point by North London rivals Arsenal.
And the chance to not only play with some of the best players in the world, but also in Europe's premier club competition was a significant factor as the 24-year-old made his desire to move to Madrid clear.
Bale was criticised by Spurs' coach Andre Villas-Boas on Wednesday after he refused to attend training last week in a bid to force the move through and chairman Daniel Levy admitted the player's wishes had played a crucial role.
"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," he said.
"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."
As a result of his rebellion Bale hasn't played at all since Spurs' pre-season friendly against Swindon Town on July 16 as he first complained of a gluteal injury and then a foot problem which kept him out of the English side's impressive start to their league campaign.
However, he has still been called up by Chris Coleman for Wales' upcoming World Cup qualifiers against Macedonia and Serbia.
The move could also spark a flurry of transfer deadline day departures from the Bernabeu with Kaka set to re-join AC Milan on a free transfer.
The Brazilian has long been surplus to requirements given the wealth of attacking midfield options on offer to coach Carlo Ancelotti and the former World Player of the Year is desperate to secure first-team football ahead of the World Cup in his homeland next year.
And Mesut Ozil could also be sold to recoup some of the funds spent on Bale with Arsenal said to be strongly interested in the €45 million rated German international.
Ozil insisted he was ready to stay and fight for his place in the side following the arrival of Isco and now Bale last week, but he was surprisingly left on the bench by Ancelotti for Sunday's 3-1 win over Athletic Bilbao.