The Scotland international has been a long-term target for Spurs, and the north London club have scouted the youngster on a number of occasions.
Villas-Boas is braced for a string of summer bids for Bale from Europe's biggest clubs - Real Madrid, Bayern Munich and Manchester United are three of the teams thought to be considering a move for the Welshman.
Tottenham hope a £130,000-a-week contract offer will be enough to persaude Bale to stay at White Hart Lane for another season at least, but Forrest has been identified as a possible replacement for the Spurs winger.
Forrest would cost a fraction of that fee - possibly up to £8m. The Scotland star has appeared in the Champions League for Neil Lennon's side this season, and is viewed as one of the Scottish Premier League's most promising young players.
Arsenal, Norwich and Wigan have all watched the Celtic man, but Spurs could offer David Bentley or Bongani Khumalo in part-exchange for the 21-year-old. And a player-plus-cash fee could be enough to tempt Celtic to part with Forrest.
Bale scored again on Thursday evening, as Tottenham chalked up a 3-0 victory over Inter Milan in the Europa League.
The Wales international has been in unstoppable form since the turn of the year - and is the club's current top scorer with 21 goals in all competitions so far.