Napoli see Garnacho as a replacement for Khvicha Kvaratskhelia, who is joining PSG in a €75m deal.
However, BBC Sport says their first attempt for Garnacho has been rejected by United.
It's understood the offer was for £40m, which falls well below United's valuation.
Indeed, United are not actively seeking to sell the Argentina international, though will listen to any crazy offer for the attacker this month.