The Welshman scored twice and provided an assist in Tottenham's 4-0 win over Burnley on Sunday.
It was a spectacular return to form for Bale, who had been inconsistent since arriving on loan from Real last summer.
And Los Blancos are hoping they can cash in if Bale's form continues.
According to the Express, Real are desperate to offload Bale's 600,000-a-week wages as well as to bolster their finances for the upcoming transfer window.
Much will depend on Bale's final few months in north London, with Spurs pushing to qualify in the top-four.
Jose Mourinho's side are also into the League Cup final and Europa League last-16.