The 24-year-old completed a world-record €100 million move to the Santiago Bernabeu on Sunday, having previously failed to show up to training for the north London club during protracted negotiations.
Afraid of having an unhappy player on their books, Tottenham finally concluded a deal for Bale a day before the deadline, and Villas-Boas says the club were put under heavy pressure to sell by both the player and Madrid.
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“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.”
Tottenham chairman Daniel Levy had previously stated that the club had no intention of selling the two-time PFA Player of the Year, but felt he would not be committed had they kept him at White Hart Lane against his will.
“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,” read a statement from the Spurs chief following Bale's departure.
Villas-Boas, meanwhile, also went on to criticise the scheduling of the transfer window, and believes it should close before the season starts and therefore ridding clubs of any uncertainty.
“The clubs like it as it is but the coaches are of the opinion that the market should be closed earlier,” he added.
“The coaches need time to prepare their teams for the season. Most coaches would like it to be open less time and close earlier; obviously we will have to have further meetings.”