The Italy international has been heavily linked with a move away from Sardinia this summer after an impressive campaign in Serie A.
Gazzetta dello Sport journalist Nicolo Schira has reported that the 22-year-old will be watched by an Old Trafford representative during this weekend's league match at the Stadio San Paolo.
United are said to be interested in making a move for the central midfielder this summer as Ole Gunnar Solskjaer plans a summer overhaul of his squad.
Napoli defender Kalidou Koulibaly is also understood to be a target, but the Italian club's president Aurelio De Laurentiis is not willing to sell him for less than his €150 million (£129 million) asking price.
Barella has made 31 appearances for his club this season, scoring one goal and providing three assists.
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