The England international, capped 24 times, injured his knee in the Champions League loss to Barcelona on 23 February.
He played 70 minutes at the StubHub Center, which followed 45 minutes against Lens on 22 July – a match in which he scored in a 1-1 draw – as well as 45 minutes in Thursday’s 2-1 win over the MLS All-Stars.
“[He gives us] power and penetration – he has it all,” added Wenger. “That’s why I would like him to get to the level he can go to.
“Maybe he doesn’t completely believe himself how good he can be and that’s the final step for him.
Arsenal conclude their pre-season schedule with a friendly against Manchester City in Gothenburg on Sunday. A week later they begin the new Premier League season against Liverpool at Emirates Stadium.