The England captain is being considered as the successor to Karim Benzema at the Bernabeu, according to El Desmarque.
Benzema could be sold at the end of the season to make way for fresh blood in the squad, but Mbappe remains the La Liga giants’ top target.
However, PSG could demand as much as €300 million for the 20-year-old after setting a €220m price tag for Neymar over the summer.
That price would be out of reach for Real Madrid and they are therefore considering other options.
Kane is at the top of that list and could leave Spurs if they fail to qualify for the Champions League this season.
The coaching team in Madrid are said to be fans of the 26-year-old and believe he is capable of adapting to life in the Spanish capital.
According to the report, Kane’s valuation could be around €200m, a fee that could be considered by Real president Florentino Perez in his search for a striker who can guarantee goals to his side.
Kane has started this season with five goals in nine Premier League games, but Mauricio Pochettino’s side have struggled so far.
Tottenham are seventh in the table on 12 points after nine games.
Zinedine Zidane’s Madrid side have also come in for criticism early in the season and suffered a shock 1-0 loss to Mallorca on Saturday.
That result saw them fall to second place in the La Liga standings, one point behind rivals Barcelona.
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