The Wales international finally completed a protracted switch to the Bernabeu over the weekend for a world-record fee of £85.3million.
Spurs boss Villas-Boas conceded it was the will of the player which proved decisive.
Speaking after a meeting of top coaches at UEFA headquarters in Switzerland, the Portuguese told The Telegraph: "The pressure we were subjected to by the player has been intense.
"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."
The summer transfer window closed on Monday, more than two weeks after the opening Premier League fixtures.
Villas-Boas does not believe that is a healthy situation, adding: "The clubs like it as it is but the coaches are of the opinion that the market should be closed earlier.
"The coaches need time to prepare their teams for the season. Most coaches would like it to be open less time and closed earlier; obviously we will have to have further meetings."
Bale is now planning to live up to his billing in the Spanish capital although he has to wait until September 14 for his Madrid debut when Real travel to Villarreal in a game you can watch live on Sky Sports 2.
Asked about the great players Real have had in the past, Bale said: "There have been the likes of (Zinedine) Zidane and obviously the previous ones, which were a bit before my time, like (Ferenc) Puskas and (Alfredo) Di Stefano.
"There are lot of players who have been unbelievable here, world class and legends of the game. Hopefully, I can become the player I want to be, and hopefully be in that league."
Bale will be managed by Carlo Ancelotti at his new club and said: "I think he's a great manager, he's won at every single club he's been at, I think.
"I've spoken to other players who have had him as manager and it's been nothing short of he's the best manager they've ever had. So obviously that was a factor in me coming, working with a great like himself.
"As for the squad, it's an absolutely amazing squad, I've got to work hard to get in the starting XI. There are world-class players here and I obviously came here to be a part of it and to try and get into the starting XI and win trophies."