Barella, a box-to-box powerhouse, has scored one goal for Cagliari in 20 Serie A games this season - with his good performances earning him four Italy caps.
Chelsea coach Maurizio Sarri sees in the midfielder a heir to Cesc Fabregas, who signed for Monaco this month, with his No2 Gianfranco Zola reportedly calling the player multiple times.
But Gazetta dello Sport report that the Cagliari starlet has now caught attention of the Red Devils, who are "very lively" monitoring his situation.
Barella would reportedly find it "impossible to say no" to United if they came knocking on Cagliari's door.
Some Italian newspapers compare the starlet to Liverpool legend Steven Gerrard.
And Inter Milan were thought to be also interested in Barella's services, but might miss out on the Italian having struggled to comply with Financial Fair Play rules.
Cagliari have slapped a £43m price tag on their star, but would like to keep Barella until the end of the season.
The player would most likely favour such an outcome as his wife is six-month pregnant.
Meanwhile, Antonio Valencia might be nearing his Old Trafford exit this month, with Newcastle and Fulham interested in United's captain.
The 33-year-old has featured in just five Premier League games this season, and is set to become a free agent in the summer.
