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Ryan, who is a boyhood Arsenal fan, joined the Gunners on loan from Brighton until the end of the season in January.
The goalkeeper has made just three appearances for Arsenal in that time as he has mainly served as the back-up to first-choice Bernd Leno.
However, Ryan has been solid when called upon and impressed club staff at the training ground.
The Australian has integrated very well into the squad, forming close relationships with a number of players, and his chances of staying at Arsenal have increased as the season has gone on.
Arsenal did not include an option-to-buy clause in Ryan’s loan deal in January so would have to negotiate a transfer with Brighton.
The goalkeeper, however, has just one year left on his deal at the Amex and has slipped down the pecking order there, which should make negotiations easier.
Ryan tweeted a farewell message to Arsenal on Monday, but added that he looked “forward to continuing the relationships well into the future”.
Ryan wrote: “Have had a blast with this last chapter experiencing all that’s been involved in representing Arsenal.
“Have learnt so much and enjoyed every minute of it.
“To the many great people around the club whom I’ve had the pleasure of associating with, thank you for making me feel welcome and making the transition into the family a very easy one, I look forward to continuing the relationships well into the future. Excited for what the future will bring. #thankyou"