According to the Daily Mirror, the Red Devils want to sign the Senegalese-born French footballer and they could spend as much as £50m on the player.
Manchester United manager Ole Gunnar Solskjaer is targeting new players in the January transfer window as he bids to boost his squad for the rest of the season.
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Solskjaer is targeting the addition of midfielders and forwards.
Boubakary Soumare is one of the midfield transfer targets on the transfer shortlist at Manchester United.
The 20-year-old is a key player for Champions League campaigners Lille and he was involved in all the six Group H games as his team finished in last position on the table.
Lille finished their campaign with a solitary point from the six games in Group H behind Dutch champions Ajax Amsterdam, Premier League side Chelsea and group leaders Valencia.
Lille were unable to qualify for the Europa League knockout stage after they were beaten to it by Ajax Amsterdam via finishing third on the table.
Boubakary Soumare has made 22 appearances in all competitions in this campaign.