Mourinho is said to be keen on strengthening his squad during the January window, with two full-backs and a creative midfielder said to be on his radar.
Moura has previously been linked with a switch to the Premier League and according to The Sun, United will look to make a deal happen before the end of the month.
The 25-year-old has not started a single game in Ligue 1 or Champions League this season, and the Brazilian international is expected to leave the Parc des Princes.
However, PSG will allegedly only contemplate a loan agreement providing that United or any interested club are prepared to pay £40m for a permanent switch in the summer.
Since joining the French giants in 2013, Moura has scored 46 goals in 207 appearances in all competitions.