The Welsh ace has joined the Spanish giants for £85.3million and will sign a six-year deal upon completion of a medical.
Bale released a statement confirming the move and thanking Spurs for his successful spell in North London.
"I have had six very happy years at Tottenham but it's the right time to say goodbye," read the statement.
"We've had some special times together over the years and I've loved every minute of it.
"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.
"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.
"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."