Neymar's future has been the subject of growing debate throughout the season amid speculation that he is unhappy in the French capital, despite only joining last summer.
Real Madrid are favourites for the former Barcelona player's signature, but Spanish outlet Don Balon claims that United hope to join the race with a big-money bid for the Brazilian.
PSG are unwilling to sell their star asset and would likely command a fee in excess of the world-record £200m they paid for him one year ago, but United are thought to be considering meeting that valuation.
The report goes on to claim that the Red Devils would offer the 26-year-old £650,000 a week, making him the world's highest-paid footballer.
United boss Jose Mourinho is expected to make sweeping changes to his squad this summer in an effort to close the gap to Manchester City, although he has previously suggested that he would not bring in any more attacking players.