The three-time Premier League winner took over at Old Trafford last summer and led the club to the EFL Cup and Europa League during his first season at the helm.
The Ligue 1 outfit are known to be long-time admirers of Mourinho, while incumbent PSG boss Unai Emery's position has been called into question at times this season despite the club's fine form both domestically and in Europe.
Mourinho also publicly praised PSG last month and the French club's owners are thought to see the Portuguese boss as their best bet to achieve their ultimate goal of winning the Champions League.
However, The Guardian reports that United have no concerns that Mourinho will leave the club and hope that he will remain in charge for more than a decade.
Mourinho has never lasted for longer than three-and-a-half years at any of his previous clubs, although he has suggested in the past that he would be willing to stay at Old Trafford for a significant period of time.
United currently sit second in the Premier League table, although they are eight points adrift of leaders Manchester City.