The Ivory Coast international, who arrived at Old Trafford from Villarreal in 2016, has made just 11 Premier League appearances this season.
According to The Sun, Gunners boss Unai Emery has been a fan of the 25-year-old since coming up against him while leading Sevilla in La Liga.
The Emirates outfit will begin talks over a fee at £20 million, with United looking likely to make a loss on a player they brought in for £30 million three years ago.
Bailly is about to enter the final year of his United contract, leaving him within the price range available for Emery to bolster his defence for next season.
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