The 19-year-old was initially signed on loan from Monaco last summer but PSG agreed to pay £166m to acquire the France international on a permanent basis.
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 month.
Nonetheless, UEFA has stated that the Qatari-owned club's finances would remain under close scrutiny.
According to The Mirror, Man United have been monitoring the situation and are now prepared to move for Mbappe, with Anthony Martial potentially utilised as a makeweight in the deal.
Mbappe is currently on World Cup duty with Bleus and scored the winner in the Group C encounter with Peru, a goal which made him France's youngest ever goalscorer in the competition.