The Dutchman will now miss the Gunners' Champions League last 16 second leg against Barcelona at the Nou Camp, with Arsenal already expected to be without fellow forward Theo Walcott through injury.
Arsenal take a 2-1 advantage to Spain on March 8.
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“Unfortunately, we had bad news on the scan and he will be out for the next three weeks for sure,” Wenger told Arsenal's official website.
“It could be longer. The shortest delay we can imagine is three weeks [so] he is definitely out for Barcelona."
Van Persie was injured when scoring the equaliser against Birmingham City, which the Blues went on to win 2-1.
Wenger did have better news to report about midfielder Alex Song.
“Song has a knee problem as well. From his scan we had good news, it is only blood from a kick on the outside of his knee. He has a good chance for Barcelona."
Arsenal host Leyton Orient in an FA Cup Fifth Round replay at Emirates Stadium on Wednesday evening, with goalkeeper Wojciech Szczesny and defender Laurent Koscielny - at fault for Birmingham's winning goal at Wembley on Sunday - set to be rested.
“Koscielny has a tight hamstring. There is a little bit of fatigue in there as well so we will give him a breather," said Wenger.
“[Manuel] Almunia needs games so he will play but Wojciech is alright and will be on the bench.
