The Welshman completed a record-breaking £85.3m move to the Santiago Bernabeu on Sunday, having been in talks over the switch for the duration of the summer.
Despite the club's unwillingness to sell their star man, Villas-Boas has cited Bale's determination to move to Spain as the reason for his departure.
"The pressure we were subjected to by the player has been intense," Villas-Boas told The Telegraph.
"The influence of Real Madrid has been very big and you get to a point where you have to consider if it's feasible not to accept an offer and whether you can press on with a player in that state of mind.
"Obviously, we should be happy with the transfer fee that was paid. Players want to win championships and I believe that with Gareth that is within reach. At the end of the day, it's up to a player to decide if he wants to stay or wants to go."
Bale signed a six-year deal with Madrid.