Bale's Spurs career came to an end on Sunday after the two clubs agreed a deal for the 24-year-old.
The Wales international scored 55 goals for the club in 203 appearances, including 21 goals in the Premier League last season.
The former Southampton player admitted that the lure of the Liga club was too much resist, but was quick to pay tribute to everyone involved at Tottenham after his departure was announced.
"I have had six very happy years at Tottenham but it's the right time to say goodbye," a statement from Bale read on the club's official website.
"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."
Bale has signed a six-year contract for Carlo Ancelotti's side and will be unveiled at the Santiago Bernabeu on Monday.
