Van Persie’s contract with the Gunners runs out next summer and he says he will not sign a new deal.
But Wenger has called for the Netherlands striker’s future to be sorted out before the start of the Premier League season on 18 August.
“He has one year left and is in demand but our desire is to keep him,” said the Arsenal boss.
“However, we will do what is in the best interests of the club. If you have a decoder in your head you can decode that.”
The Gunners have already rejected offers from Manchester City, Manchester United and Juventus for the Dutchman.
Van Persie scored 37 goals last season, 30 of them in the Premier League, and won both the PFA and Football Writers’ Association Player of the Year awards.
But, following a third-place finish in last season’s Premier League, the 28-year-old rejected a new contract at Arsenal and questioned the club’s ambition.
Wenger, speaking in Malaysia at the start of Arsenal’s pre-season tour, said: “We are very, very ambitious. You know that. Not only very ambitious, but also very proud of the way we run our club.
“Van Persie is a world-class striker and I am a big supporter of him. I have supported him through his whole career.”
Wenger wants Van Persie’s future resolved quickly to avoid a repeat of last summer’s drawn out transfer sagas involving Samir Nasri and Cesc Fabregas.
Nasri was sold to Manchester City and Fabregas to Barcelona late in the transfer window after a summer of speculation.
“The official transfer deadline is 31 August but then we can create an internal deadline,” Wenger said.
Asked if that was before the new season kicks off, he replied: “Of course. I believe the transfer market should stop before the season starts anyway.”
Gunners midfielder Aaron Ramsey had also stated his desire for Van Persie to stay.
“He had a brilliant season last year and is a big player for us, so we are just hopeful that he will stay,” said Welshman Ramsey, who is on Team GB Olympic duty.
“We don’t know anything that goes on. We hope that he stays at Arsenal.”