The winger previously suggested playing under previous Madrid manager Zinedine Zidane would be a "dream" but the Frenchman has since stepped down as boss.
With six goals so far at the World Cup, Kane looks a dead cert to take home a golden boot, and after over 100 Premier League goals in just five seasons, he is said to be an "excellent replacement".
Gab Marcotti discusses the latest developments in the Conte-Sarri saga and what's keeping the former Napoli manager from moving to Stamford Bridge.
Can Eden Hazard lead Belgium all the way to the final of the World Cup at the expense of France?
"Does Real make me dream less without Zidane?"
"For now, it's the World Cup". 'Zidane is someone special, it's true, but I think Real is everyone's dream.
According to the same story, Real Madrid are determined to land their man this summer and Perez has already made a decision to give the number seven shirt to the Belgium worldwide.
And Abramovich is said to be ready to sell Hazard as he is considering selling the club, although he has not yet received an offer.
According to Catalan daily Mundo Deportivo, the Italian has communicated to the Chelsea hierarchy that he does not want Hazard and Willian to walk out of the club in the ongoing window.
The Portugal global scored two goals against Juventus in Real's 4-1 win in the 2017 Champions League final and earlier this season struck a jaw-dropping bicycle kick against the Italians in a 3-0 win in a quarter-final first-leg game in Turin which prompted Juve supporters to give him a standing ovation.
Dermot Corrigan is a Madrid-based football writer who covers La Liga and the Spain national team for ESPN FC.