The Spanish giants are ready to break the world-record transfer fee for Spurs' Welsh winger.
Real are already understood to have tabled an £85million offer for Bale and could go even higher, although White Hart Lane chairman Daniel Levy appears determined to hang onto his star man.
However, Real legend and director of football Zidane believes unsettled Bale should be given the opportunity to discuss a possible move.
Speaking from the US, where Real are preparing for Friday's friendly against Los Angeles Galaxy, Zidane said: "If he's expressed a desire to join Madrid, then Tottenham should give him permission to speak with us.
"The chance to play for Real Madrid might only come around once in a player's lifetime - and it's understandable Gareth doesn't want to miss out on it."
Bale is due to report back at Tottenham on Wednesday, although a muscle injury means he is unlikely to train and will probably not be involved in the weekend friendly against Monaco.
The 24-year-old is said to be frustrated by Levy's stance on the pursuit by Real, and Zidane added: "It's not unusual when a player knows that Real Madrid are interested in them, that they get excited.
"There is nothing more prestigious for a player to wear than the white of Real Madrid."
Carlo Ancelotti's side have already spent an estimated €70million (£61million) this summer signing Isco, Asier Illarramendi, Dani Carvajal and Casemiro, but Zidane says the club is not put off by the numbers involved in any possible deal for Bale.
"This is modern football. You have to pay to get the best players in Europe," said the Frenchman.
"Last season, Gareth was on the same level as Cristiano Ronaldo and Lionel Messi. The best players can have not so great games but change a game with a moment of brilliance.
"Those are the sort of players that win you Champions League and league titles."
Zidane also believes Bale can improve his game at the Bernabeu.
"He is already one of the finest players in Europe and he has everything needed to become the best player in the world," he added.
"It's a bit of a cliche but it's true - you become a better player playing with better players."
Meanwhile, Sergio Ramos has adopted a more ambiguous tone with regards to the possibility of Real signing Bale.
"I think that the price of the player is dictated by the market," said Ramos.
"It's not my job to decide whether we should sign Bale or someone else. It's important for Madrid to have a large number of young players and Spanish players in the team. The mix of youth and experience makes for a good project. The president and the other directors will decide whether or not we sign Bale, not me."