The 19-year-old made the move to Barcelona last summer from Croatian outfit NK Kustosija, and he has represented the Catalan outfit's second team on 24 occasions this term, scoring three times in the process.
Faye is yet to make his first-team debut for the La Liga champions, but he was on the bench for the team's recent league contest with Mallorca.
According to the Daily Mail, Man United have been monitoring the teenager's form this term, with the Red Devils considering making a move when the summer market opens for business.
The report claims that the 20-time English champions had scouts in attendance to watch Faye make his international debut for Senegal on Friday, when he scored a long-range effort against Gabon.
Barcelona are believed to have rejected offers of around £13m from both Nice and Lens in January, but the interest is growing ahead of the summer transfer window.
Bayer Leverkusen and Inter Milan are allegedly both keen on Faye, with Barcelona expected to listen to offers of around £25m for the youngster due to their ongoing financial problems.
Man United reportedly believe that their biggest threat when it comes to signing the defender is from Leverkusen, with the Bundesliga outfit pushing for a deal.
Faye has scored three times in his last five league appearances for Barcelona B, and he is expected to continue with the second team for the rest of the campaign.
Man United are expected to sign at least one centre-back this summer, with Raphael Varane and Jonny Evans both only having contracts until the end of the campaign.
The Red Devils are thought to be keen to
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