The 63-year-old has been with the London club since 1996, and has overseen three Premier League title wins and four FA Cup successes in that time.
The silverware has dried up of late, with the Gunners' last trophy coming in 2005, but Wenger has ensured UEFA Champions League qualification for 16 consecutive seasons.
And, following speculation regarding the Frenchman's future, Gazidis has confirmed that the club's board are fully behind their manager.
"We think we have got a fantastic manager. We hope that he wants to do what he is doing for the long term. I believe he does," he said. "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 into place." 
Gazidis also insisted that the club have ambitions beyond simply qualifying for Europe, and underlined his desire to see trophies brought to the Emirates Stadium.
He added: "We want to be competing at the top of the game and in order to do that you have to be in the Champions League, so we are pleased to qualified, or at least, for the qualification games. 
"But it is not ultimately where we want to be with moving the club forward. We want to be a club that is competing at the very top end of the game and that means competing to win the Premier League and competing to win the Champions League."