Sardinian newspaper La Nuova claim both clubs have been keeping tabs on the 20-year-old attacking midfielder, who they describe as the 'jewel' of Massimo Rastelli's squad.
Scouts were present on Sunday as Barella started behind the front two in Cagliari's 1-0 away defeat to Fiorentina and impressed - winning almost all his personal duels with Borja Valero.
The article states that Liverpool and Chelsea have watched Barella in two away games recently, with Cagliari's previous trip being their 2-1 win away at Crotone.
Most of the top clubs in Serie A are also keen on the youngster, an Italy under-19 international, with AC Milan, Juventus and Inter all having been linked in the past.
Roma have also been associated with a move for Barella, who is under contract until 2021, but Cagliari president Tommaso Giulini recently spoke to the player's agent to tell him the club have no intention of selling this summer.