The 24-year-old was in lethal form for the Premier League side in 2017, scoring 39 goals in the calendar year to not only set a new domestic record but also rank as the leading scorer for club and country among Europe's top leagues.
Kane's form has caught the eye of Los Blancos, who have been linked with a bid of at least £150m in recent weeks, and according to Diario Gol, the proposed deal is already some way down the line.
The publication claims that the England international has even gone so far as to select his squad number and wants to have the number 10 shirt, currently worn by ex-Spurs midfielder Luka Modric.
Madrid president Florentino Perez is said to be aware that his Spurs counterpart Daniel Levy will not entertain the thought of Kane leaving unless an offer of at least £150m is on the table, but he is optimistic of getting a deal done this summer and will try to include Gareth Bale as a makeweight in the deal to lower the cash outlay.
Meanwhile, Spurs have been linked with offering Kane improved terms worth £200,000 a week, although the player himself has since denied that any contract talks have taken place.