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."