Speculation has been rife since the season ended that the Welshman - both PFA and Football Writers' player of the year in 2013 - could be heading out of White Hart Lane.
A number of sides have shown an interest in the 23-year-old, most notably Spanish giants Real Madrid.
But Spurs manager Villas-Boas, speaking to Marca, said: "We have the pledge of the chairman that Bale is not going to leave the club."
Bale, who was the leading Tottenham player during 2012/13, scored 31 goals for club and country last season.
Real have not hidden their desire to sign Bale, with club ambassador Zinedine Zidane making a number of flattering comments about the Wales international.
Yet Levy pledged not to let go of the north London club's top stars this summer, and Villas-Boas sees no reason why the Spurs chairman will go back on his word.
Real's partnership with Tottenham - forged last year following the sale of Luka Modric to the nine-time European champions - has not stopped Zidane flirting with the idea of Bale moving to Santiago Bernabeu this summer.
In May the former France star said Bale, who has recently returned from his holidays in the United States, would fit in well at the Spanish club and last weekend he intimated that Bale could speed up a transfer if he asked to leave Tottenham.
Franco Baldini's appointment as director of football has raised hopes that Spurs will start moving to bring in new additions soon, but it is understood that the club are unlikely to announce any signings in the next couple of weeks.
Villas-Boas will be hoping that Baldini can land at least one big-name signing this summer after missing out on Joao Moutinho and Leandro Damiao in the last two windows.
A striker remains a priority for the Portuguese because of Tottenham's reliance on Bale last season.
Barcelona striker David Villa is a leading target, but it is thought that Tottenham's interest in Brazil striker Damiao has cooled.
Fellow Brazilians Paulinho and Bernard are also reported to be on Villas-Boas' radar as he looks to secure the top-four spot that eluded the north London club last term.