The 24-year-old striker is set to return for rehabilitation in his native Holland as he recovers from a thigh problem - the latest fitness setback of what has been an injury-plagued season.
Van Persie had started the campaign in fine form, before being sidelined by a knee problem suffered on international duty during October and then a muscular injury put him out again until the new year.
Wenger brought the forward back into the side for the first leg of the Carling Cup semi-final against Tottenham at the Emirates Stadium on January 9, but substituted him at the interval.
Following a bout of 'flu, medical tests have revealed a "little recurrence" of Van Persie's thigh injury, which will see him spend probably the best part of a month on the sidelines once again, and possibly become a doubt for the Champions League clash with AC Milan on February 20.
Wenger knows it is "difficult" for the player, who has been limited to just 13 starts but netted seven goals.
"We do not think it will be a long-term problem," said the Arsenal manager.
"Maybe we made a mistake to play him in the Carling Cup, because we wanted to rush him a little bit back after the muscular problems and were not cautious enough maybe on the night.
"This time we will have to be very patient."
Wenger added: "When a player has waited for so long and is ready to play, you are under pressure a bit from him as well because he feels he can play.
"Also, when you have been out for a few months, you need some games to get back to your best. You want to give him those games as soon as possible.
"At the end of the day, I do not feel we made the right decision to play him on the night."
The Arsenal boss has sanctioned a trip back to Holland as part of Van Persie's "treatment and rehabilitation".
Wenger said: "A football club is not a hospital. It is made for players who play, not for those who cannot play or are sick.
"It is very depressing for them to come in every day and see their partners going out and playing, when he cannot.
"Sometimes it is better to get them away for 10 days where they have a completely different environment. It makes them feel better."
