The British Broadcasting Corp, without saying where it got the information, put the fee for Bale at more than the record figure Real paid Manchester United for Cristiano Ronaldo four years ago.
The Welshman has agreed a six-year deal with Real believed to be worth 100 million euros (4.2 billion baht) a year, which would make Bale the world's most expensive player, it said.
However, media in Spain insisted that Real will only pay 91 million euros, three million less than Los Blancos paid for Cristiano Ronaldo back in 2009.
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.
"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."
Real fans and executives will have to wait to find out if the investment in Bale, who scored 55 goals in 203 appearances for Spurs since May 2007, works out in footballing and financial terms, said former HSBC banker Eugenio Martinez, who sought to stand against Florentino Perez in a 2009 presidential election.
"There's no doubt Bale is an extraordinary player," Martinez said in an e-mail. "Only time will tell if he's expensive or cheap."
The record nine-time European champion will try to recoup as much of the fee as possible through commercial deals and jersey sales, said Liam Wong, an analyst at London-based Brand Finance, which ranks Real as football's third-biggest brand.
"Real Madrid is a commercial machine and will do its best to turn Bale into a global star," he said.
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 24-year-old Bale 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.