Ramsey has entered the final year of his current deal at the Emirates Stadium and will be free to discuss terms with foreign clubs from January onwards.
Barcelona and Serie A outfit Lazio are among the teams to have been linked with the Wales international, while there is also understood to be serious interest from a number of Chinese Super League clubs.
Arsenal's Premier League rivals Chelsea, Manchester United and Liverpool are all thought to be monitoring Ramsey's situation too, but The Mirror reports that the Gunners are confident of keeping him in North London.
Ramsey is reportedly willing to bide his time in order to improve his bargaining position as he holds out for an offer in excess of £250,000 per week.
Arsenal gave Mesut Ozil a new deal worth £350,000 a week in January, and Ramsey believes that he also deserves a significant pay rise following a 10-year stay at the club.