The Daily Star says the midfielder is available on a free transfer after his contract at Manchester City expired at the end of the season.
Everton will have to pay City £1m if Barry signs a permanent deal owing to the loan agreement with the Premier League champions.
But they face stiff competition from the Gunners who can offer the veteran midfielder one last chance for Champions League football.
Barry is also being tempted by a move Stateside to finish his career off in the MLS.
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