The Tottenham star, 25, is said to be their top target as they search for a new man to lead the line after a difficult season so far.
And Spanish publication AS say they have been quoted what would comfortably be a world-record fee by Spurs.
That would dwarf the current record set by Paris Saint-Germain when they paid Barcelona £200m to sign Neymar, 26, in 2016.
As a result, the report claims they might turn to more realistic options, with Bayern Munich star Robert Lewandowski, 30, and PSG’s Edinson Cavani, 31, high on their list of targets.
Kane, though, remains their dream target and Real have cash to spare after selling Cristiano Ronaldo, 33, to Juventus for £100m last summer.
They have struggled since the Portuguese superstar left, most notably in front of goal.
Santiago Solari’s side are languishing in fifth in the La Liga table and are a whopping 10 points behind leaders and bitter rivals Barcelona after 18 games.
They have scored just 26 times in the league, while Barca have almost twice that with 50 strikes in 2018/19.
Karim Benzema, 31, is Real’s top scorer in La Liga but the Frenchman only has seven goals.
Former Spurs star Gareth Bale, 29, is next on the list with just four.
And they might turn to north London to solve their problems once more having signed Luka Modric, 33, and Bale from Spurs in the past.
They are also said to be extremely keen on bringing Christian Eriksen, 26, to Spain, with Modric thought to be weighing up a switch to Inter Milan.
Chelsea’s Eden Hazard, 28, remains a long-term target and he is yet to make a decision on his long-term future.
They have also been linked with River Plate’s Exequiel Palacios, 20.
The midfielder impressed as he helped the Argentine giants to the Copa Libertadores title this year and revealed he will discuss the move with his family before making a decision.