The Sun says Kane, who starred in Tottenham's 1-1 draw at the Bernabeu on Tuesday, is already their No 1 target.
Now Spurs face a season-long battle to keep hold of their prized asset — a fight they lost when winger Gareth Bale moved to Spain for a then world-record of £86 million in 2013.
And it is clear only a fee close to the £198m world record Paris Saint-Germain paid Barcelona for Neymar in the summer could tempt the Londoners to sell.
Tottenham have no intention of letting their 24-year-old superstar leave and the England striker insists he is happy at the club.
But the appeal of playing for 12-time European Cup winners Real is impossible to turn down, especially for players with ambitions of winning the big trophies.