The Armenian joined the Giallorossi on a season-long loan in August and he has since endured a mixed campaign where he has made just 20 appearances because of injuries.
Despite this, he has been effective with six goals and four assists in all competitions and the Serie A side are said to be keen on signing him on a long-term contract.
According to Corriere dello Sport, Roma want to spend around £13.7m plus add-ons on the 31-year-old winger but that is currently far off from the Gunners' £22m price tag.
The north London side are unlikely to come down from that price for Mkhitaryan and reports have previously emerged that manager
could reinstate the attacker into his plans for next season.
Arteta is understood to be an admirer of the former Borussia Dortmund man and the Armenian could be offered a lifeline during the opening weeks of the 2020/21 season.