United boss Jose Mourinho regards the Brazilian as the best young prospect in Europe and will urge Ed Woodward, the club's executive vice chairman, to broker a deal in January.
It is understood that Bordeaux are asking for more than £55m, and with president Stephane Martin insisting the 20-year-old will not be allowed to leave this winter, United are prepared to let him stay at the Matmut Atlantique stadium until the summer.
Malcom has already matched last season's tally of seven goals in Ligue 1 but despite some eye-catching performancs this season the French club are languishing in 15th place.
The Red Devils are currently some way short of offering the sums suggested by Bordeaux, but Mourinho will want Woodward to be flexible with his budget after the United supremo refused to meet Inter's valuation of Ivan Perisic in the summer, a source of annoyance for the Portuguese manager.
It is believed Arsenal have also been keeping tabs on Malcom, although they are yet to have made any contact with Bordeaux.