The report goes on to claim Real are plotting a £51m move for the France worldwide, who has joined Chelsea team-mate Eden Hazard on their summer transfer wishlist.
The Belgian midfielder is back to his best after struggling last season, and is leading Chelsea's charge for the Premier League title.
Since moving to Stamford Bridge, Kante has established himself as an integral member of Antonio Conte's four-man midfield, carrying on from where he left off at the Foxes.
"But it wasn't a success because they said things like 'we have more players like you in our teams and we didn't need you'". So I kept working in my local team Suresnes, a little team in a suburb of Paris. I have been in two good teams.
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Chelsea defeated City 2-1 midweek at Stamford Bridge with a trip to Dean Court to play AFC Bournemouth to come this weekend.
At Boulogne and Caen, the midfielder made up for lost time.
The 26-year-old added: "Because I played in the lower divisions in France, to have the chance to train with players from academies made me think "ok, I need to keep learning, working and improving".
"That's the reason why I didn't go to Arsenal and went to Chelsea".
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"If he is able to, he is a top, top player".
Arriving for £30million from Leicester last summer, Kante has been a key player with title glory on the cards.
He told The Mirror: "Last season we were champions with Leicester and at the moment we are top of the league with Chelsea".
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