Wenger has repeatedly delayed a decision over whether he will remain at the club beyond the end of his current contract, which expires this summer.
Pressure on the long-serving Frenchman to step down has grown throughout the season and has reached fever pitch during an ongoing run of just one win in six Premier League games.
Arsenal have also been thrashed 10-2 on aggregate by Bayern Munich in that time, leading to regular protests from fans imploring Wenger to end his reign.
The club's hierarchy are understood to have tabled a new two-year offer for the 67-year-old but, speaking at a fans forum event ahead of this afternoon's 2-2 draw with Manchester City, Gazidis hinted that recent results could force them into a "change".
It is not thought that he was referring directly to Wenger's position, though, despite the Gunners now sitting seven points adrift of the top four.