Ramsey, 27, will see his current deal at the Emirates Stadium expire at the end of this season.
Talks between the two parties over an extension broke down last month, and it appears that the versatile midfielder is heading for a move away from the Gunners.
Liverpool and United have both been credited with an interest in the Welshman, but according to the Daily Mail, Juventus are also very much in the race.
Indeed, the report claims that the Old Lady will approach Ramsey with a contract offer at the start of 2019.
Ramsey is closing in on 350 appearances for Arsenal in all competitions following a 2008 move from Cardiff City.