Los Blancos have been frequently linked with the England frontman since last summer, but The Sun (h/t Daily Star's Callum Vurley) reported Kane has "vowed" to stick with Spurs amid rumours of a move to Spain.
It's suggested Real Madrid president Florentino Perez is keen to recruit a new batch of Galacticos at the Estadio Santiago Bernabeu, with the club disappointing in the 2017-18 campaign.
ESPN's Alex Shaw recently hinted at the danger of Kane leaving in order to pursue major title ambitions elsewhere:
Real striker Karim Benzema in particular has struggled to inspire as part of their attack this season, netting only four times in 21 La Liga appearances as Zinedine Zidane's side sit third in Spain's top flight.
Cristiano Ronaldo's recent return to form has helped make up for the lack of goals from their starting striker, although broadcaster Deji Faremi has hinted at Benzema's place being safe as long as results continue to come:
Deji Faremi @deejayfaremi
@tobiogunsakin @K1Says @iamOkon @thebusybuddies @stbanky007 Again, you must realise that Real Madrid have moved beyond the era of building individuals to the era of building a team. Benzema isn't scoring many goals, but if the team is doing well, I believe RMA will be fine...but let's not digress #BBFansChat
Kane, meanwhile, is contending for his third successive Premier League Golden Boot award and sits on 24 Premier League goals for the campaign, having scored 35 times in 38 outings across all competitions.
Benzema has as many assists this season as he does goals (eight), and Tottenham recently provided evidence to support the notion Kane is more than just a finisher himself:
Tottenham fans will be desperate to see their talisman remain in north London for years to come, with the 24-year-old having built a strong relationship with manager Mauricio Pochettino in recent years.
With a new stadium en route and the team playing its best football in modern times, there is promise around Spurs at present, but Kane may need to see trophies if he's to stay for the long term.
Real succeeded in luring former Tottenham star Gareth Bale to Madrid, and they might stand as one of the few clubs able to afford a Kane purchase, not to mention offering the temptation it would require to pull off such a deal.
Kane reportedly remains committed to the cause in north London for the time being, and with more than four years left on his contract, Spurs aren't likely to be panicking anytime soon despite Real's interest.