Spanish reports suggest the Spanish giants are ready to break the £80million world-record transfer fee set by their acquisition of Cristiano Ronaldo by offering £86m for Bale.
The 24-year-old enjoyed a phenomenal campaign last season, netting 21 times in the Premier League and winning both the PFA and FWA Player of the Year awards along with the PFA Young Player of the Year.
Despite Bale's contribution, Spurs were unable to secure a top-four place in the Premier League, meaning they are unable to offer their prized asset UEFA Champions League football.
The change in betting odds is in contrast to news coming out of White Hart Lane, with Tottenham insisting Bale is not for sale and talks over a new contract are taking place.
Sky Bet Football Trader Chris Spicer said: "The majority of bets throughout the summer have been on Bale remaining at Tottenham, but we've seen a sudden shift in interest in him leaving and we now offer 7/4 on him staying at Spurs.
"Reports suggest that Real Madrid have now made their move and that Bale is unhappy that it has been swiftly rebuffed.
"Our punters now seem convinced Real will get their man and Spurs will have to settle for their sizeable compensation."
Odds subject to change.