The Sun says United scouts have repeatedly checked on the attacking midfielder - including in Bordeaux's 1-1 draw with Marseille last Sunday.
United boss Jose Mourinho has been on 20-year-old Malcom's trail since last season, and had him watched in February.
The Brazilian has hit five goals and set up four more for his French club this season.
Malcom signed a contract extension until 2021 with the Ligue 1 club in September.
He hinted earlier this year that he would be open to a move to the Premier League.