The Brazil international has been heavily linked with a move away from Ligue 1 in recent months, despite only joining the French giants last summer.
Neymar, 26, is said to be open to quitting PSG this summer to join a more competitive league.
And according to Spanish news outlet Don Balon, who are well known for their outlandish transfer claims, United and Real Madrid are battling to land the winger this summer.
However, they say that PSG are not willing to negotiate a deal with Real, they are only open to talking to United.
Don Balon add that the French outfit will sell Neymar to the Red Devils, if they pay a staggering £346m (€400m).
PSG feel they would rather let Neymar join United at the end of the season over Real if the attacker is determined to leave Paris.
Neymar joined PSG last summer in a world-record £200m deal from Barcelona after the Ligue 1 champions met his release clause.
The attacker is currently sidelined with a foot injury but the Brazil star is hoping to be fit for the World Cup in Russia this summer.
United are back in action tomorrow night when Jose Mourinho’s men travel to Bournemouth in the Premier League.