According to the BBC, the 24-year-old has always dreamed of playing for the La Liga giants, who are yet to register a bid for the Premier League star.
Recent reports in Spain have suggested a transfer offer for the winger is close, while Manchester United look set to be priced out of any bidding war by Madrid.
Spurs chairman Daniel Levy is still hoping their star player can be persuaded to stay however, and is even said to be ready to refuse a record-breaking £85m bid.
Bale scored 26 goals last season in all competitions, with his stand-out season coming in 2010, scoring a hat-trick against Inter Milan.
Tottenham have been busy on their pre-season tour this month, arriving back in London on Sunday where Bale missed out on playing in either of the club's matches in the Asia Trophy because of a muscle problem.