The Dutchman sustained ligament damage while scoring the leveller against Birmingham City in the final of the League Cup, which the Gunners eventually lost.
Former Feyenoord hitman Van Persie did not last long enough to see the Blues take the lead for a second time in the game, as he had to be taken off after 69 minutes.
Having already ruled the 27-year-old out of the upcoming Champions League second leg against Barcelona, Wenger said: “I am not a specialist but he overstretched his knee at the back when he made the volley [to score on Sunday].
“He got kicked or blocked by the defender and overstretched when he hit the ball. In fact we lost Van Persie after 30 minutes, we tried to keep him going in the second half but he was injured.
“It is a ligament at the back of the knee, a tear in the ligament. It is three to four weeks, I take the short delay because I am an optimist but it could be four. As well it is the international break during that period so whether it is three or four [weeks], for us it is the same.
“We have two weeks to play now and then there is a two-week break because of international games. We have to hope it is only four; he cannot play before the international break, that is for sure. Let’s hope he can play after the international break.
“I listen to what the medical people say to me and you never know, he can have a setback. You can never guarantee.”
Arsenal’s first game after the international break is the home fixture with Blackburn in April.