The Frenchman has just 12 months to run on his current deal and has come under pressure from sections of the club's fans after another season without silverware, which has seen the Gunners' trophy drought extend to eight years.
Gazidis was coy on the details of an extension and even if one was already on the table, but the former MLS deputy commissioner believes Wenger is the right man to make Arsenal title challengers again.
"We think we have got a fantastic manager," Gazidis told reporters. "We hope that he wants to do what he is doing for the long term. I believe he does. I think he is still ambitious, still driven and sees the potential of the club as he looks forward and I think he is very excited by that.
"We have a great relationship and he has a great relationship with the board as well. So, quietly and at the right time I think we will make an announcement on that when things are all put in place.
"This is going to happen very quietly behind closed doors, privately and then there will be an announcement. We have got a lot of confidence in Arsene that he is the right person to take the club forward and I think he will want to do that."
Gazidis also dismissed claims that any lingering doubt over Wenger's drive for the job could affect Arsenal's appeal in the transfer market, and feels the "managerial musical chairs" at other clubs has halted business from being completed.
"If you are asking me, is that an issue with players, the answer is 'No,'" Gazidis said. "It is far more an issue with players signing at other clubs than it is at Arsenal. If players have that on their issue list, I think that’s a mark in our favor, not against us.
"I think players that are concerned about uncertainty probably think about Arsenal as the most certain place they could be in the world of football. This is a club that has had remarkable consistency in terms of its manager, its football philosophy, its direction and the consistent support from the board.
"It has been quite a slow start [to the transfer window] and I think that’s because there was such a managerial musical chairs going on that everybody is waiting for that to settle down.
"I think clubs, agents and players are, which means it has been quite an unusual beginning to the transfer window. I imagine once most of the managerial appointments settle in a little bit, we will see that starting to pick up."