Bale completed a world-record move from Tottenham to the Spanish club in September following the conclusion of a lengthy close-season transfer saga.
But, while Perez has insisted the winger was desperate to join the club he supported as a child, he has also revealed that the Premier League champions were preparing a rival bid.
"Tottenham did not want to sell and we respected that. Then, when we found out that Manchester United would be making an offer, we made our own," Perez told Cadena SER.
"Since his childhood he had the dream of playing for Real Madrid and he asked us to make it possible for him to come."
Despite the size of the Welshman's price-tag, which Barcelona coach Gerardo Martino recently claimed showed a "lack of respect", Perez also claimed that Bale was actually a "cheap" purchase.
"Bale was cheap," he continued. "The signing of Gareth Bale is the model of Real Madrid, signing the best player in the world at this moment, and that is what I say.
"All the great players of Real Madrid have been cheap. I will not say the amount, but we must not focus too much on the purchase price.
"Some numbers that have come out I find difficult to understand and, if I were a journalist, I would not have reported them."
Bale is currently suffering with a thigh injury that forced him out of the starting XI for Real's clash with Getafe on Sunday.
He will miss Wednesday's trip to Elche as a result and is a doubt for the Madrid derby on Saturday.
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