The Welshman is out of contract at the end of the season and with the campaign having now moved into 2019 he is free to discuss a pre-contract with overseas clubs.
Real Madrid, PSG, Bayern Munich and Juventus have all been linked with approaches for the playmaker, but it is understood that the Italians are Ramsey's first-choice destination.
The 28-year-old faces a big decision over his future, with Arsenal having pulled out of talks over a new deal back in October due to a combination of footballing and financial reasons.
Ramsey is into his 11th year on the books at Arsenal, having racked up 353 appearances and 61 goals since making the move to the English capital from Cardiff City back in 2008.
The classy midfielder has struggled to secure regular football under Unai Emery, starting just eight Premier League games so far this season and he is believed to have prioritised a bosman move to Turin when the season ends.