Wenger this month celebrated 18 years as manager at Arsenal, who need look no further than Manchester United and life after Alex Ferguson to see how difficult it is to replace such a pivotal figure at a football club.
The Frenchman, who is 65 next week, signed a three-year contract extension in May after ending a nine-year trophy drought by winning the FA Cup.
Gazidis said: “So much of the attention on a football club is around an individual and that’s very understandable at Arsenal because we have a giant who’s managing us.
“Arsene has been a fantastic driver and has put the club in a great, great position. The biggest challenge we’re going to face as a club is that, when that transition from Arsene to the next manager of our football club happens — and I don’t know when that’s going to be — that we come through that strongly.”
Gazidis, who has been in post for six years, is optimistic for the future and wants Arsenal to join the world’s elite.
He added: “I wouldn’t be doing this job if I didn’t believe we could compete at the very top level. I know Arsene feels exactly the same way.
“Where we are currently is off the shoulder of the world’s top teams. I think the best teams in the world are in Europe and you think about the great names in world football like Real Madrid, Barcelona and Bayern Munich — I don’t think we’re at that level yet but that’s where we’re aiming to get to.
“It’s one thing to say that and it’s another thing to be able to deliver it.”
He added: “It has been incredibly frustrating not being able to be where we want to be these last few years, so winning the FA Cup was a fantastic moment,” Gazidis added.
“(But) I don’t come out of that feeling satisfied like we’ve achieved what we need to achieve. The hunger for success is immediate. The moment you get away from that euphoria of the moment, the next question is, ’How do we have more of that?’ That’s what everybody at this club is focused on. There’s no satisfaction in it — it’s what do we do now.”
Wenger must first contend with a growing injury list after Tomas Rosicky joined Mesut Ozil and Laurent Koscielny in reporting a problem while on international duty, while Danny Welbeck went off against Estonia for England.
Rosicky, who did not feature in Czech Republic’s win over Kazakhstan on Monday after suffering a calf problem against Turkey, said: “I couldn’t do any sharp movement, couldn’t sprint.
“I’ll come back to London and we’ll see. The doctor said I have something swollen between the muscles.”
Rosicky has made just one start and five substitute appearances this term, but the prognosis over other key men remains to be fully determined.
Ozil (knee) could be out until early 2015, but Koscielny and Welbeck could play against Hull on Saturday.
Koscielny has a minor Achilles problem, while England boss Roy Hodgson was sure Welbeck’s ankle injury was not serious. There has been good news for Wenger this international break as Theo Walcott returned to training on Monday following his long-term knee injury.