Wenger is calling time on an Arsenal career that goes back 22 years. He will no longer be the club's manager come the summer.
But Gazidis did not give away too many details about how the club would approach the succession during his press conference on Friday at the Emirates Stadium.
When asked about how Gazidis would go about searching for Wenger's successor, he said: "The process begins today but I want to keep that in house. I don't want to be making public announcements about that.
"I don't underestimate the challenge of that. You don't find a replacement for Arsene Wenger, you find a new path forward."