The Frenchman's near 22-year reign in charge of the club is over, with Arsenal yet to confirm who will be taking charge for the 2018-19 season.
Moving forward, Arteta is considered as what Arsenal are looking for in the wake of Wenger.
Frontrunners for Wenger's Arsenal job emerge
The former French Gunner has been linked with the vacant position at the Emirates but Vieira, now head coach of MLS side New York City FC, has apparently been left "disappointed" after being contacted by Arsenal.
Gazidis is working his way through a preferred list of candidates with head of football relations Raul Sanllehi and head of recruitment Sven Mislintat.
It has been mooted that Santi Cazorla could be offered a new player-coaching deal, with a role like Michael Carrick's in his last playing season at Manchester United.
Arteta spent five years as an Arsenal player at the end of his career, making 149 appearances for the club across all competitions.
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