As noted by Sky Sports, a story from Onda Cero suggested the two clubs had staged some preliminary talks when they met in the UEFA Champions League earlier in the week, with Spurs said to have quoted a price of £200 million for their star forward.
But according to Sky Sports, Tottenham have no knowledge of such negotiations: "Tottenham's stance over Kane remains that their star striker is not for sale at any price."
With Real Madrid hosting Spurs in the Champions League, links between the forward and the European champions have been ramped up.
According to Jose Felix Diaz of Marca, Real have been monitoring the development of the England international after his incredible rise with Tottenham. However, he added they are unlikely to sign a new striker in the January market.
"All eyes were on the Tottenham Hotspur star as he faced Real in midweek but Madrid bosses were not dazzled by his performance and made no attempt to open negotiations for the striker when meeting their North London counterparts," added Diaz.
The man himself has hinted that he would be happy to stay with his current side for the remainder of his career too, per the Daily Mirror (h/t Bleacher Report UK):
Given the way in which he's progressed at Tottenham under the watchful eye of manager Mauricio Pochettino, that isn't a massive surprise.
Indeed, when he first burst on to the scene at Spurs, Kane appeared to be a limited but hard-working forward. Now he's got so many different strings to his bow, as he's able to bully defenders, link play and find intelligent positions on the field.
Crucially, his goalscoring record has been sensational for Tottenham and England. As noted by BT Sport Football, ahead of their meeting in the Champions League, Kane has been just as prolific as Cristiano Ronaldo during the calendar year:
According to Diaz, there have been some talks at the Santiago Bernabeu about adding a new forward to Zinedine Zidane's squad. And as one of the game's rising stars, Kane will always be linked with the likes of Real.
At the moment, Karim Benzema is the only natural focal point in the setup; Ronaldo is still more effective when given a free role off the left or as part of a front two, whereas Borja Mayoral is still potential at this point. Another natural goal-getter would be a big boost to Los Blancos.
But, as relayed by Harry Sherlock of Goal, given the progress they have made lately, Spurs will be desperate to keep Kane around:
It'll be intriguing to see how the 24-year-old fares when he takes on Madrid in the return fixture at Wembley, as a goal or two in a winning performance will send the speculation into overdrive.
For the time being, it would be a massive surprise if Spurs were involved in any talks with Madrid whatsoever, as Pochettino is attempting to build a team capable of winning the biggest prizes. Granted, Tottenham are far from a one-man outfit, but Kane offers the x-factor that's so key in clinching those types of cups.