The 18-year-old is hot property after scoring 19 goals from 32 games this season.
His goal tally is more than United and Arsenal stars Wayne Rooney, Olivier Giroud, Lucas Perez and Anthony Martial.
Most expensive Premier League squads fielded this season
Fri, March 17, 2017
Which Premier League side has fielded the most expensive starting XI this season?
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20. Sunderland: £45.90m against Watford in Gameweek 17
Kylian Mbappe is France football's latest star
Kylian Mbappe has been in stunning form for Monaco this season
Real Madrid have also been backed to make a move for him.
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But French newspaper Le Parisien claim that France football's latest star is wanted by Paris Saint-Germain.
Still reeling from their 6-1 Champions League defeat to Barcelona, manager Unai Emery is plotting a summer spending spree.
Monaco would rather sell him abroad than to their rivals but PSG believe £80million could ensure they land their man.
Arsenal will have to spend more than the £42m they paid to sign Mesut Ozil
It means that, should United and Arsenal wish to sign him, they may have to match the Ligue 1 giants' offer.
Any deal would make him Arsenal's highest transfer of all time.
Should he sign for United, however, he will be their second most expensive purchase in history behind Paul Pogba.