Van Persie scored 37 times for the Gunners last season but he could leave the Emirates Stadium this summer if he fails to agree an extension to his current contract, which expires at the end of next season.
Talks over Van Persie's contract ended without agreement last month and Manchester City have vowed to bid for the striker if he becomes available.
Van Persie, who is on international duty with Holland at Euro 2012, has described another one of his suitors, Juventus, as a "great team", further fuelling suggestions he could quit London.
But despite the uncertainty over their skipper's future, Gazidis says the Gunners will not be pressured into making a declaration on the matter soon.
"Robin sat down with us at the end of the season and we had a good discussion," Gazidis told Arsenal's 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."
City signed Samir Nasri from Arsenal under similar circumstances last season when the midfielder had 12 months left on his contract.
The Frenchman went on to win the Barclays Premier League with his new club, and City could offer Van Persie a bumper contract way above what he currently earns with Arsenal.
Van Persie's value to Arsenal is clear. The Dutchman's 30 Barclays Premier League goals almost single-handedly rescued them from major failure last season as they fought back from a woeful start to finish third.
There were calls for Arsene Wenger's head when Arsenal lost three of their opening five games - one of which was an 8-2 drubbing at Manchester United - but Gazidis is glad the club kept faith with their manager of 16 years.
He said: "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. That is not what happened."
Regardless of Arsenal's turnaround, Gazidis still thinks the club underachieved last season and is determined to see them challenging for trophies again next term.
"None of this is cause for popping champagne corks at Arsenal," the 47-year-old said. "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."
Last summer Wenger dallied on replacements for Nasri, Cesc Fabregas and Gael Clichy, with Andre Santos, Per Mertesacker and Mikel Arteta all arriving on deadline day.
The new arrivals were seen as inferior to the men they replaced, but Arteta went on to have an outstanding season.
So far this summer Arsenal have already signed Germany striker Lukas Podolski and are keen on recruiting France midfielder Yann M'Vila, although he has stated he will wait until after Euro 2012 before deciding his future.
Gazidis hinted that Wenger will try to move as quickly as possible to make new additions to his squad.
"We always try and conduct our activity as early as we can," he added. "The key for us is not the volume of signings, but that they add something to the squad.
"That is something Arsene thinks extremely carefully about and for all of the pressure he is placed under, he tends to make good decisions."