Sport says Real have now drawn up a seven-man transfer shortlist, which includes some of the tournament's biggest stars.
Los Blancos are said to be 'closely following' the World Cup and analysing players who fit into their philosophy.
It is said that they are particularly targeting a left winger, a midfielder and a forward. It is expected that as many as four of their current squad will depart the Bernabeu in next summer's window.
For that reason, they have now drawn up a shortlist with Kane the headline. Also included are Jude Bellingham (Borussia Dortmund), Alphonso Davies (Bayern Munich), Enzo Fernandez (Benfica), Youri Tielemans (Leicester City), Youssoufa Moukoko (Borussia Dortmund) and Stanislav Lobotka (Napoli).