The 27-year old Wales international is entering the final 12 months of his deal at the Emirates and could leave on a free transfer next summer.
Ramsey joined the Gunners in 2008 from hometown club Cardiff City and has scored 36 goals in 234 Premier League appearances.
Former Arsenal defender Martin Keown has said Arsenal should be 'desperate' to keep hold of Ramsey.
The Wales midfielder scored 11 goals in 32 appearances for the club last season.
Speaking following the draw in Pasadena, Ramsey said he was excited by the appointment of the former Sevilla and Paris St-Germain boss Unai Emery to replace Arsene Wenger.
"I haven't spoken to him [Emery] at all yet, but I think it's an exciting time for the club and I can't wait to get back there now," said Ramsey.
Wales manager Ryan Giggs has insisted Emery's arrival could improve the player who finished the game against Mexico as captain following Ashley Williams' first-half injury.
"He's only played under Arsene at Arsenal and I look at it positively that a new manager could come in and improve him, improve already a quality player," said Giggs
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