The 19-year-old has enjoyed a stratospheric rise to the upper echelons of football following his exploits with Monaco in 2016-17, with PSG last season and with France at the World Cup.
PSG had considered selling Mbappe this summer if they were found in breach of UEFA's Financial Fair Play regulations, but the European governing body cleared the Parisians earlier this summer.
Nonetheless, UEFA has stated that the Qatari-owned club's finances would remain under close scrutiny.
According to Spanish outlet Don Balon, the Red Devils are prepared to part ways with Paul Pogba for a fee in the region of £150m and launch a £300m bid for Mbappe.
Mbappe, who lifted the World Cup with Les Bleus on Sunday, is thought to have cost PSG £120m to sign from Monaco, making him the second-most expensive player in the world behind teammate Neymar.