Addressing a question-and-answer session with the club's supporters before the 2-2 draw, Gazidis said results and form under Wenger over the past six weeks had been "disappointing".
The Mirror has claimed a sporting director could be appointed this summer to work alongside Wenger, who will have to accept the board 's ecommendations before he signs a new two-year £8m-a-year contract that is on the table.
There is also speculation that Arsenal will look to overhaul their scouting set-up.
Arsenal fans protesting against Wenger handed leaflets to fellow supporters with a large van carrying anti-Wenger quotes also parking up outside the Emirates Stadium.
Wenger has increasingly split supporters in recent weeks, with six defeats in nine games having left the north London club out of the Champions League and sixth in the Premier League ahead of City's visit.
The Frenchman's contract expires in the summer and he is yet to announce whether he will sign a new deal or leave the club he has managed for over 20 years.
Two protest marches took place ahead of recent home games, while a plane was flown over the Hawthorns calling for Wenger to leave, although that message was offset by a pro-Wenger banner also taking to the skies as West Brom beat Arsenal 3-1.
Almost £4,000 has been raised through a crowdfunding campaign, with some of the funds used to decorate a van with quotes from Wenger and Arsenal chief executive Ivan Gazidis.
Read the full article on eurosport.co.uk: Arsenal chief Gazidis hints at big changes if Wenger remains