Diario Gol says Florentino sees Kane as a natural replacement for Karim Benzema, who was quietly put up for sale before Christmas. Any deal is unlikely to happen before the World Cup, however.
In the meantime, Florentino is exploring how to ferry Kane away from London and is considering a swap proposal - which would involve two Spurs old boys.
One of Gareth Bale or Luka Modric would be offered to Tottenham this summer, along with Mateo Kovacic, to drive down the valuation of Tottenham for Kane.
Florentino knows Spurs are keen to re-sign Bale, having rejected a €50m offer from chairman Daniel Levy before Christmas.
Indeed, it's been suggested the president would be willing to send all three Real players to Spurs in exchange for Kane if it convinced Levy to do business.