The 28-year-old's current contract expires at the end of next season and speculation is mounting that he could leave the club this summer amidst interest from Manchester City.
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"Robin sat down with us at the end of the season and we had a good discussion," Gazidis told the club's official website.
"What we agreed at that meeting was that we would keep all the discussions we had over the summer to ourselves and make announcements when it is the right time.
"We have to respect the fact we have agreed to keep that among ourselves. Robin is clearly focused on the Euros at the moment and we wish him well, and at the right time we will make the right announcements."
The north London outfit suffered a poor start to last season, slipping as low as 17th in the Premier League and enduring an 8-2 hammering against Manchester United at Old Trafford.However, Arsenal turned their fortunes around to clinch third place and Champions League qualification for next season following a late resurgence.
And Gazidis expressed delight that the club had remained patient with manager Arsene Wenger but is now targeting a Premier League title challenge.
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"This football club was placed under terrific pressure at times and I think for a different club, there would have been the possibility that we would begin to fall apart, panic and make decisions that would have been bad," the 47-year-old added.
"That is not what happened. None of this is cause for popping champagne corks at Arsenal while it is a creditable performance it is not achieving our ambition.
“We want to win the Premier League. We want to compete to win the Champions League.
“That is what we are planning for during the course of the summer and everything we do is geared towards that.”