The 21-year-old is under contract with the Serie A club until 2021, but he has a £22 million release clause which the Gunners are looking to exploit.
The London Evening Standard has claimed that Arsenal have gone cool on signing Jean Michael Seri from Nice and believe Pellegrini represents better value for money.
A product of Roma’s youth academy, Pellegrini made his senior debut for the club in 2015 as a second-half substitute in a 1–0 league win over Cesena.
He joined Sassuolo for £1.1 million in June that year, although the deal included a buy-back clause of around £9m.
He went on to score 11 goals in 54 appearances for Sassuolo, before agreeing a move back to Roma in June 2017.
The midfielder played 37 times for Roma in all competitions, scoring three goals.
Pellegrini has been capped six times by Italy and played against England in a friendly earlier this year.
Liverpool have also been linked with Pellegrini in the past, but they appear to have switched their attention to other targets.