The Sun reports that the Spanish giants could include Portuguese left-back Fabio Coentrao in a £50million deal for Bale.
Current Spurs boss Andre Villas-Boas tried to sign Coentrao for Chelsea last year.
Villas-Boas also wanted to include the 24-year-old in the deal that took Luka Modric from White Hart Lane to Real Madrid.
Bale signed a new five-year contract with Spurs in August.
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Summer 2016 is set to be a very exciting time for football fans, as this is when the long awaited Euro 2016 tournament kicks off.
Since Guus Hiddink arrived at the helm, Chelsea have improved considerably after a tricky start to the campaign.