The 18-year-old full-back has made 33 first-team appearances and won three Wales caps since breaking into the Bluebirds first team in August last year.
The two clubs reached agreement over what is reported to be a £2million deal last week and the player underwent a medical over the weekend.
Cardiff boss Dave Jones admits there was no way the club "could stand in the way" of his dream move.
"He is a young prospect - we wish him all the best," Jones told Sky Sports News.
"You can't deny the lad the opportunity, he's done really well and broken into the full international team.
"What we said to him was go and prove to everybody the potential we all thought you had.
"He's a very raw boy, he's still got a lot to learn.
"If he breaks into Tottenham's first team this season it would be fantastic for the kid."
And Jones believes Juande Ramos has signed a player every bit as promising as Bale, who joined from Coca-Cola Championship side Southampton in the summer.
He added: "He's very athletic, he's got great pace and can pick out a pass. He will get stronger and stronger.
"He's still not fully developed and I think Tottenham know they've bought somebody who they're going to have to nature along, just like we were trying to do here.
"Maybe they will have to be a little bit patient with him, because he is still learning his trade.
"I feel they've bought another good prospect like Gareth Bale - he's just as good and capable on the right-hand side."