"If you are very positive you would say [he will be back in] April," manager Arsene Wenger told the Premier League club's website on Tuesday.
"If you're a bit cautious you say May. It is virtually the rest of the season."
Netherlands start their World Cup campaign in South Africa with a Group E tie against Denmark on June 14.
Van Persie, 26, had an operation on his right ankle after being hurt in a friendly international against Italy last month and the initial forecast was that he would be sidelined for four to five months.
"When you have such a big surgery there is one thing that is important - the opinion of the surgeon. Will he be the player he was before? We had a clear answer - yes," said Wenger.
"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," added Wenger.
Arsenal are seeking compensation from the Dutch Football Federation as they believe the KNVB initially misdiagnosed the injury.
The KNVB, though, denies any wrongdoing.
Van Persie was in top form before the injury, notching eight goals in 11 matches for Arsenal.
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