The newspaper reports Madrid made contact with Spurs in December to get a sense of how much the Premier League side would want for the England global.
If Los Blancos do match the £312m fee, it would easily surpass the current record transfer set in 2016, when Paris Saint-Germain paid £200m to sign Neymar from Barcelona.
The likes of Bayern Munich's Robert Lewandowksi and PSG's Edinson Cavani could be available for the right price, but both center-forwards over 30-years and Los Blancos want a lasting solution.
The Spurs man is a reliable goalscorer having bagged 155 goals over the last five seasons in all competitions, while Ronaldo has netted 213 times in the same time period.
Tottenham have also rejected bids from the Spanish giants for Kane's teammate Christian Eriksen as the Danish worldwide is arguably playing the best football in his career this season.
The Champions League holders believes Kane can perfectly fit in the void Ronaldo left at the club.
Given Kane's lethal goalscoring form, Spurs will not want to part ways with their prized asset, hence the massive asking price, and it certainly would be a shock if Real were to meet it.