The Times says Gazidis has received an offer from AC Milan, who have been taken over by his friend Paul Singer, but has yet to inform the club of his decision.
Arsenal last week denied the 53-year-old had handed in his notice but the North London club's powerbrokers and manager Emery are growing increasingly frustrated at the situation.
Gazidis has helped restructure the Premier League side's management over recent years.
He has overseen the departure of Arsene Wenger and made a series of appointments to go with Emery's arrival, most notably head of recruitment Sven Mislintat.
Arsenal's staff have privately accepted the chief will leave, but they are desperate for clarification.