Former United boss Sir Alex Ferguson tried to sign the Brazilian in 2012 but Paris St-Germain stole the deal at the last minute with a monster bid.
Louis van Gaal is on the lookout for a pacey winger and French paper L'Equipe report that Moura, 23, is on his radar.
However, United face a battle for his signature with a number of Premier League clubs reportedly interested.
Moura has scored three goals in 12 appearances this season and put pen-to-paper on a new four-year contract in the summer.
"I love being in Paris," Moura said after extending his deal.
"I want to make history. I feel that I can win the Champions League with PSG.