The Spanish giants have been tracking Kane, 24, as a possible replacement for Karim Benzema.
Their season was derailed on Saturday when they were thrashed 3-0 by Barcelona and signings are likely to come in January.
But that is unlikely to be Kane, who says he is dreaming of eclipsing Alan Shearer by becoming the Premier League's all-time record scorer.
The Tottenham striker took his top-flight total to 93 with a hat-trick at Burnley on Saturday and is chasing more goals against Southampton on Boxing Day.
One goal will see Kane shatter Shearer's record for Premier League goals in a calendar year – set when he hit 36 for Blackburn in 1995.
But the Spurs frontman's long-term aim is to chase down Shearer's tally of 260 scored for Rovers and Newcastle.
Kane said: "Of course, you dream about it. But I'd be silly to look that far ahead. I've still got a long way to go.
"A lot can happen in football. I hope I can stay fit, stay sharp and we'll see what happens."
Nobody has ever got close to Shearer's tally, which does not include the 23 top-flight goals he scored for Southampton before the Premier League was formed in 1992.
Wayne Rooney is the closest, with 208 goals for Everton and Manchester United, but at 32 is running out of time to catch his former England team-mate.
Kane added: "Maybe if I get to 200 and I've got a few more years left, I'll start to think about it. For now I'm concentrating on each game.
"I've had a great 2017 and I hope I can improve in 2018 and get even better."