Spanish newspaper AS report that Real Madrid have made him their top summer target but also claim a Premier League team has already made a world record £96million bid – and had it rejected.
Mbappe has broken onto the world scene this year with his stunning performances in Ligue 1 and the Champions League.
The 18-year-old scored in both legs as Monaco dumped Manchester City out of the Champions League at the last-16 stage.
Kylian Mbappe is on Real Madrid's radar for this summer
Kylian Mbappe has scored 11 in his last nine games
"Mbappe is a footballer that at one point is going to end up being a worldwide star," AS quote a source as saying.
"He reminds us a lot of Ronaldo, when at that age dazzled at PSV Eindhoven, or Henry when he wanted to bring Madrid the next Trezeguet."
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Mbappe has scored a sensational 11 in his last nine games and has 12 Ligue 1 goals in 11 starts and 11 substitute appearances.
Monaco are keen to keep hold of him and believe their Champions League progress gives them a chance although with the bids already reaching the astronomical point it is hard to see them holding out.
Jose Mourinho has had him and Monaco team-mate Tiemoue Bakayoko heavily scouted this season, but appears to have lost out on the midfielder with Chelsea now the front-runners for his signature.
United also paid big for Anthony Martial from Monaco just over 18 months ago, in a deal that could reach over £50m, but Mbappe looks to be on course to pull in am even higher fee.