Moura headed to the French capital in 2012 when his club side Sao Paulo were offered £40m, significantly more than Sir Alex Ferguson was prepared to hand over at the time.
Ferguson famously blasted the big-money move, claiming that "the game's gone mad", but United could now be about to offer a similar sum for the 24-year-old to lure him away from the Parc des Princes.
According to Sport Witness, Jose Mourinho has earmarked Moura as an ideal midfield option for next season and will rival Liverpool for his signature.
The Reds are said to be prepared to offer £30m for the Brazil international, who finished the season with 19 goals and six assists for PSG across 53 appearances in all competitions.
It is claimed that contact has already been made between United and the Parisians, with the Red Devils confident of adding the winger to their squad once the transfer window opens on July 1.