He was injured in a collision with Aston Villa defender Antonio Luna in Arsenal's opening day Premier League home defeat on Saturday which ruled him out of their Champions League qualifier at Fenerbahce on Wednesday.
British media had reported that the 20-year-old could be out for up to six months but Wenger allayed those concerns.
"It will be at least six weeks out with a posterior cruciate problem... a ligament that is stretched. We don't think it will need surgery," the Arsenal manager told a news conference ahead of the playoff first leg in Turkey on Wednesday.
The Arsenal manager has found himself under early pressure following Saturday's disappointing 3-1 reverse against Villa which followed a difficult close season in the transfer market.
The north London club made a failed bid to sign Uruguay striker Luis Suarez from Liverpool and were rebuked by Newcastle United manager Alan Pardew on Monday for lacking respect after making an offer for French midfielder Yohan Cabaye.
Arsenal's bid came before Newcastle's game at Manchester City which they lost 4-0 with Cabaye left out of the team.
"I do not know whether it is fair or unfair," Wenger told reporters. "I just do what is best for this football club."
Arsenal, who finished fourth last season to clinch a Champions League playoff spot, have so far signed only Yaya Sanogo on a free transfer from Auxerre and Wenger is under pressure from disgruntled fans to sign some quality players.
The Frenchman said he will try to reinforce his squad but emphasised that he has faith in the players he already has.
"As always, we do what we think is right," he was quoted as saying on the club's official website (www.arsenal.com). "When we think we have the right players, we do it.
"We were for years restricted and we competed with exceptional quality. This year we did not lose any players - we just lost two players with injury - but we will try to strengthen the squad and make the right decisions until the end.
"I repeat many, many times that the transfer window should be over before the season starts because it is a little bit destabilising once the season has started - to be there talking about what is not really important in football."
(Reporting by Toby Davis; editing by Ken Ferris)
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