The 24-year-old missed their Asia Cup friendlies in Hong Kong due to a muscle strain but that has not stopped him being the centre of attention off the pitch.
Reports in the Spanish press - most notably in sports daily Marca - claim Bale wants to join Real and that their president Florentino Perez has sanctioned a bid in excess of £80m.
Tottenham have refused to comment on the speculation but manager Andre Villas Boas revealed on the Friday that the club and Bale's agent are in 'ongoing' talks about a new contract.
Real have arrived in Los Angeles for their tour of the United States along with Perez, who has broken the world transfer record three times whilst in charge at the Bernabeu.
He signed Luis Figo for £37m from arch-rivals Barcelona in 2000, bought Zinedine Zidane for £46m a year later and then set the current record of £80m with the capture of Cristiano Ronaldo from Manchester United in 2009.
Real Madrid play LA Galaxy in their first game in the International Champions Cup on Thursday and will then face either Everton or Juventus.
Meanwhile, Bale is for the first time this summer odds-on with Sky Bet to leave Spurs.
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.