The Wales star has agreed a £300,000 per week, six-year deal after sealing an £85.3m (100m euros) move which eclipses the £80m Real paid Manchester United for Cristiano Ronaldo in 2009.
"I have had six very happy years at Tottenham but it's the right time to say goodbye," said 24-year-old Bale.
"We've had some special times together and I've loved every minute of it."
Bale will have a medical and then be presented to the Real Madrid fans at the Bernabeu at 12:00 BST on Monday before speaking at a media conference.
He added: "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 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 joined Spurs as a left-back for £10m from Southampton in 2007 and scored 26 goals last season as he was named both the Professional Footballers' Association's and Football Writers' player of the year.
Bale's season-by-season Premier League stats
2012-13: Played 33, goals 21, assists four
2011-12: Played 36, goals nine, assists 10
2010-11: Played 30, goals seven, assists one
2009-10: Played 23, goals three, assists five
2008-09: Played 16, goals 0, assists 0
2007-08: Played eight, goals two, assists 0
His representatives told Spurs at the end of July that he wanted to speak to the Spanish giants.
Real Madrid coach Carlo Ancelotti's claim at the start of August that they were "in talks to find a solution" to the transfer angered Tottenham manager Andre Villas-Boas, but the clubs have now reached a settlement.
After his side's 1-0 loss to Arsenal, Villas-Boas said: "Bale is a wonderful player. He is going to join Real Madrid so we wish him all the best.
"He left us with some great memories from last season."
Bale, who becomes Real Madrid's fifth signing of the summer, paid tribute to both Southampton and Tottenham for developing his career.
He said: "I am well aware 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.
"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."
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