The Tottenham star has been heavily linked with a move to the Bernabeu in recent months.
Real boss Zinedine Zidane and club president Florentino Perez are both said to be big admirers of the England international, 24.
And according to Spanish transfer outlet Don Balon, Kane is keen to join the La Liga champions.
They say the striker has told Mauricio Pochettino he wants to leave Tottenham should he decides to quit the north London club.
Kane doesn’t want to join a Premier League rival but he’s open to a move abroad.
Don Balon add that Real talisman Cristiano Ronaldo is desperate for the Spanish giants to land Kane next summer.
“Harry Kane is keen to join Real Madrid”
Kane has been vocal on his future, revealing he could end up playing abroad.
He said: “I would never say no or yes. You’d not rule out anything.
“It’s not something I definitely want to do, but you never rule anything out.
“I’m very happy here (at Tottenham), but we’ll see what happens.”
Kane has faced Real twice in the last month, with Tottenham picking up a point at the Bernabeu before beating Zidane’s men at Wembley last week.