The Wales international has been linked with a move away from White Hart Lane after scoring 11 goals in 21 Premier League appearances this season.
Real Madrid are among the clubs to have been touted as a possible destination for the 23-year-old winger.
But Villas-Boas is hoping that the north Londoners will be able to convince him to stay at White Hart Lane by sealing a top-four finish.
“Can we keep him? Hopefully yes,” Villas-Boas told reporters.
“It’s difficult for us to have a player of this dimension playing so well, but in the same way it is difficult for other teams – with the economic situation as it is – to buy a player of this dimension.
“At the moment we are very fortunate to have him. I think we can reach our objectives with him at this level, and if we reach our objectives we can hopefully continue to have Gareth in our club.
“Would Champions League qualification secure his future? I think so. This is where most of the clubs want to be, and where most of the players want to play.
“It’s part of our objectives so hopefully achieving it means we can attract better players and continue to hold on to our best players.”
Villas-Boas was gushing with praise for Bale when asked to assess his contribution to Spurs so far this season.
“He’s having a tremendous year, he’s such an outstanding player all round,” he added. “We saw yesterday his performance was outstanding once again [for Wales] and I think he’s really enjoying his football.
“Is he the best in the league? It’s very difficult, there are lots of players in contention for an award like that. I think he’s certainly amongst the five best players, for sure.
“Players who perform at this level always get the attention – it doesn’t mean that the club doesn’t foresee a continuous future with Gareth.
“I think it’s intimately related to the objectives that we achieve for the season. He’s a player we rate very highly and we aren’t willing to let go.”
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