Barella made his professional debut three years ago and is widely considered as one of the top young players in Italian football.
The 21-year-old has attracted interests from several Serie A clubs including Juventus, Inter and Roma.
But, according to Gazzetta dello Sport, Arsenal are also in the race to get the young star’s signature and have sent a scout to keep track of Barella’s progress in Cagliari.
However, The Seria A club are not willing to let the midfielder leave during the transfer window in January, and reportedly value him at €50 million (£44m).
Arsenal may also face competition from Liverpool, with reports claiming that Jurgen Klopp’s side are confident of winning the race for Barella.
The Gunners are in the market looking for a potential replacement for Ramsey after the club withdrew their offer of a new contract to the midfielder.
The Welsh international is now expected to leave the Emirates on a free transfer at the end of the season, which will leave a wide gap in Unai Emery’s midfield.