The 21-year-old midfielder was a key part of the side that battled through to the Champions League semi-finals in 2017-18.
And amid a move to revamp the Roma squad ahead of the coming season, Pellegrini might be on the verge of a switch to the Premier League.
Arsenal, who revealed Unai Emery as Arsene Wenger's successor on Tuesday, are confirmed admirers of the youngster.
And while the Gunners are not prepared to shell out £27 million [€30m/$35m] to meet Pellegrini's release clause, they will try to tempt Roma into selling with a bid worth around £22m.
The 33-year-old would be available for roughly €30m, with Pellegrini and Kevin Strootman widely tipped to leave the Serie A club.