Foot Mercato says United intermediaries have now made Bouaddi's camp and Lille aware of their interest.
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Arsenal have already contacted his representatives to finalise a winter transfer for around €45m.
The midfielder, a breakout star for Lille at just 18 years old, has made 18 appearances this season and is one of the most sought-after players on the market.
Lille, determined to keep him, have just extended his contract until 2029 and will only consider a summer sale for more than €50m.
France's Under-21 team and Morocco are also vying to secure the Franco-Moroccan gem.