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.
"It's very important for Gareth to throw his heart and soul into this move.
"He's got it all and ability-wise he is phenomenal. The only question is whether he's got the mental strength to cope with such a big transfer and the spotlight that will be immense in Spain.
"But it's a fabulous opportunity for him and the important thing is he really needs to grasp it with both hands."
Read more analysis from Gary Lineker and BBC Sport's chief football correspondent Phil McNulty
"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.
His representatives  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."
Spurs have been busy in this transfer window, signing seven players for a total outlay approaching £105m.
Brazil midfielder Paulinho has arrived from Corinthians for around £17m; forward Roberto Soldado has signed from Valencia for £26m, and Roma playmaker Erik Lamela has joined for a reported £25.7m.
Their other summer arrivals are Nacer Chadli from FC Twente for £7m, Etienne Capoue for £9m from Toulouse, Romania centre-back Vlad Chiriches from Steaua Bucharest for £8.5m, and Ajax's 21-year-old Danish midfielder Christian Eriksen for £11.5m.
Tottenham chairman Daniel Levy admitted he did not expect to have to plan for the season without Bale.
"Gareth was a player we had absolutely no intention of selling as we look to build for the future," said Levy.
"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, 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 to which we have added no fewer than seven top internationals.
"More importantly, we have an immense team spirit and a dressing room that is hungry for success."
Bale's sale could now trigger a series of deals involving other Premier League clubs before the transfer window closes at 23:00 BST on Monday.
Southampton will receive £2m as a result of the move but that figure could have been 10 times as much, had they not decided to allow Spurs to buy out the player's 15% sell-on clause in 2007.
At the time, Southampton were struggling financially and so decided to sell the clause back to Tottenham for £3m.
As part of the deal, Southampton got Spurs youth goalkeeper Tommy Forecast although he never played a senior game. So instead of an £11m windfall, Southampton will have to make do with a £2m Fifa solidarity payment, which comes their way for developing the player.
Real Madrid have broken the world transfer record to sign Tottenham forward Gareth Bale.