Wenger does not expect Van Persie back this term
The 26-year-old Dutchman suffered the injury while on international duty in November and was initially expected to return to full fitness in six weeks. But instead surgery was required, and Wenger concedes the problem will keep Van Persie sidelined for several more months.
"If you are very positive you would say [he'll be back in] April,'' Wenger told the club's website. "If you're a bit cautious you say May. It is virtually the rest of the season.
"Personally I thought 'Let's put it in my head that he will not play again this season and hope that I will have a good surprise'.''
However, Wenger is confident Van Persie will bounce back from the injury and rediscover his top form next season.
"When you have such a big surgery there is one thing that is important first of all, and that is the option of the surgeon, will he be the player he was before? We had a clear answer. 'Yes'.
"The second thing that is very important in the first stage after surgery is has he an infection or not? He hasn't. The third thing is patience. At the moment he cannot do anything, he is just sitting at home waiting until he can work again. The first two steps have been very successful.''
With such an important member of his attack gone, Wenger is expected to dip into the transfer market next month and the Frenchman admitted he would have no problem signing a player who is cup-tied for the Champions League, just as he did in signing Andrei Arshavin last January.
"Yes, I would,'' Wenger said when asked. "We are going for the domestic title and the FA Cup as well and doing that allows you to rest a player for the Champions League or play another player in the Premier League.
"Of course I would prefer the player who can play in both competitions and so it would mean the player has the quality to play in the long-term for Arsenal Football Club.
"But we have an interested eye on the transfer window and if the right player turns up then we will do it. Sometimes you think that is a bargain you cannot turn down and don't forget Arshavin got us in the Champions League last season.''