Van Persie damaged ankle ligaments playing in the Netherlands' friendly against Italy in November and he was ruled out until the spring.
The former Feyenoord ace underwent an operation earlier this month and there are no fears for the 26-year-old's long-term playing career.
It is now likely to be May before the Dutchman is ready to return, however, Wenger is keeping his fingers crossed that Van Persie may be back in April to take part in the final weeks of the Premier League season.
"It's virtually the rest of the season," admitted Wenger. "I haven't personally said that to Robin.
"Hopefully, we'll have a good surprise."