The Sun is reporting that the Red Devils are interested in bringing the Portuguese star to Old Trafford as they begin to ponder their plans for the summer transfer window.
The same report claims that Manchester United boss Jose Mourinho is a keen admirer of the the 22-year-old, who cost Monaco £13.4m from Benfica.
However, the same article states that Monaco could demand a fee as high as £70m for the winger because his current contract with Monaco does not expire until 2020.
The story goes on to claim that Manchester United are believed to have already opened talks about a swoop for the player in June.
Silva has scored 21 goals in 87 Ligue 1 games since joining Monaco from Benfica in 2014.
Manchester United are currently in sixth place in the Premier League table as they prepare for their clash with Watford at Old Trafford on Saturday.
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