It is understood Wenger attempted to sign the 32-year-old on loan last summer, only for Manchester City to block the move as they did not want Barry joining a title rival.
Barry completed the loan switch to Everton instead, where his outstanding form under Roberto Martinez has prompted talk of an England recall.
Martinez also attempted to make the move permanent in January, which would have freed up space for a loan move for former Toffees star Jack Rodwell.
City were prepared to sell Barry and allow Rodwell to return to Goodison Park on loan for the remainder of the season.
However the move broke down when Barry indicated he wanted to wait until the summer to decide his future.
Barry is out of contract with City at the end of the season, and will be allowed to move on a free transfer.
Arsenal on the other hand, have Schalke star Julian Draxler as their number one transfer target for the summer.
The Gunners held talks over a possible January move, but were unwilling to meet the asking price of £37million.
Wenger remains keen on Barry however, and is sure he can do a job for the Gunners next season.