The likes of Liverpool and Manchester United are also keen on the young defensive midfielder who has made waves for Monaco this year, but it appears a move to the Bernabau beckons for the highly-rated 22-year-old star.
The Frenchman was born in Rouen, situated in the north of France.
His father, Fernand Tchouameni, was a failed footballer and, at the time of Aurelien's birth, was an only child.
As a result of his father's failure in his own career, he played football every day with his son in the hope he would make it as a professional - and it certainly had the desired effect.
Tchouameni improved massively and was picked up by Bordeaux as a youngster.
Here he progressed through the relevant age groups and slowly started to make people sit up and take notice.
Former Chelsea and Tottenham star Gus Poyet decided he was ready for his first-team debut and played him in the Europa League and Ligue 1 at the age of just 18 during the 2018/19 campaign.
He made such an impact that the likes of Internazionale started to pay attention to this new midfielder, who was both physical and technically excellent.
His dad revealed in an interview that he saw similarities with Paul Pogba, saying: "Aurelien is a central midfielder, mentally strong, with a good ability to interpret the game and a very good technique. Compare him to a known player? He has a profile similar to that of Paul Pogba."
After his breakthrough year and halfway through the 2019/20 season, Monaco decided Tchouameni was worth a punt and they handed just over £16million to Bordeaux for his signature.
Monaco have become a club famous for their development of players, with the likes of Kylian Mbappe, Fabinho, Bernardo Silva and Thomas Lemar all honing their skills with the Ligue 1 side, before moving on.
However, his Monaco career didn't get off to the best start, as he only made three appearances, totalling 62 minutes on the pitch.
A change in manager saw Tchouameni handed a chance to impress, and the rest, as they say, is history.
The midfielder turned in some colossal performances in the Monaco engine room that helped the club finish third in Ligue 1.
Tchouameni might be a defensive midfielder player, but former Arsenal and Chelsea star Cesc Fabregas revealed last year that the 22-year-old could play anywhere.
"He can play as a No.6 alone, as a defensive midfielder, as a No.8, in a double 6," said the Spaniard.
"What he's doing now, even in a year's time, he could become a good No.10 if he wanted. He's got the power to get into the box. He needs to improve his final pass a little bit, but he's so young, and he's determined to learn.
"From what I can see, and the will that he has, he has a bright future ahead of him."
After a summer of speculation in which Chelsea and Tottenham Hotspur were linked, the Frenchman remained at Monaco, where he has continued to shine.
International recognition arrived for the midfielder, and he even played in the UEFA Nations League final against Spain, despite having made his debut for Les Bleus a month earlier.
He played the full 90 minutes as France ran out victors, and it appears as if he will have a long and successful international career.
Certainly one man who is very keen on Tchouameni is Pogba: "He's very good.
"He's not a boy, he's shown on the field that he's a man who has lots of energy.
"It's a pleasure to play beside him, I'll always be there to push him. He brings a lot. He has exceptional technique and physical qualities.
"It's nice to see young players coming into the France team and showing themselves so much. He can only progress. He's very, very young and he is the present and the future."
This is quite incredible praise from one of France's best players, and a World Cup winner as well.
Then came the links to the likes of Manchester United and Real Madrid, which led to his Monaco teammates mocking him.
He told L'Equipe: "In the dressing room, they put the cover up and laughed' 'Oh, you're leaving us already?'
"It's flattering, but I don't wake up in the morning thinking about it."
Then there was his performance against Paris Saint-Germain in March as Monaco ended the year with ten wins in 11.
Tchouameni played at the base of the midfield and turned in a masterclass against Marco Verratti, Leandro Paredes and Danilo Pereira.
This display led to numerous short YouTube videos being created to show just how good the Frenchman was against the biggest team in France.
And now he looks set for Real Madrid, which you would have to say he is more than ready for.
And given he is just 22, Tchouameni is a name that we might just all have to get used to seeing for a very long time.