After a stunning season at White Hart Lane the winger has attracted serious interest from Los Blancos, and the president has dropped the strongest hint yet of making an offer
The 23-year-old scored 21 Premier League goals last season as Spurs finished fifth in the Premier League, and he is expected to sign a new contract worth 130,000 pounds at White Hart Lane this summer.
However, Bale's agent, Jonathan Barnett, admitted on Wednesday that the attacker would consider any offer from the Liga giant.
Real Madrid striker Alvaro Morata, meanwhile, has hailed Bale as "one of the best players in the world" and backed him to be a success if he joins the club.
"He would be a good signing for the club," he told Marca. Bale is one of the best players in the world and at Real Madrid we always have the best. I am convinced that if the club want him he would be a great signing."
Last week, Real Madrid director of football Zinedine Zidane told Goal.com he rates the Wales international alongside La Liga legends Lionel Messi and Cristiano Ronaldo.
"I have been watching Tottenham for the very reason that Bale has been playing," he said. "Aside from Ronaldo and Messi, who are the top players, I must admit that he is the footballer who impressed me most."