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Gazidis confirmed last month that he was leaving the Gunners in order to join Milan, where he is due to begin work at the end of October.
The future of Ramsey remains uncertain, though, with the Welsh midfielder out of contract at the end of the season and no closer to agreeing an extension with the club he has represented for 10 years.
The agency representing Ramsey have suggested that he will leave the club either in January or next summer, and Italian newspaper Corriere dello Sport reports that Gazidis is hopeful of using his Arsenal connections to take the 27-year-old to the San Siro.
Ramsey could leave on a free transfer next summer or for a cut-price deal in January, when he is able to begin negotiations with foreign clubs anyway.