Bale, 23, collected the Football Writers' Association Footballer of the Year award on Thursday to go with the Player and Young Player of the Year awards given to him by the Professional Footballers' Association.
These accolades were in recognition of a season that has seen him score 30 times for club and country to make him one of the most prized assets in the Premier League.
Bale is contracted until 2016, but Villas-Boas would like the Welshman to extend this deal.
"It would be good for him (to sign a new contract)," said the Tottenham manager. "I think he deserves it for the season he's had. Obviously we could put to bed every single speculation that's coming out.
"It would be excellent and I think a reward for what he has achieved this season."
Bale has given the firmest hint yet that he is willing to remain at White Hart Lane, telling the Guardian: "I want to play at the highest level possible and be playing against the best teams in the world.
"I think the club is going in the right direction. All our objectives are the same, we just have to try to achieve them now."
Bale continued: "The club has grown every year and I've become a better player every year. We've had a taste of Champions League football and that's where we want to be again.
"That's where the club needs to be to get to the next level. That's what we're trying to do this season to lift the club to better things."