And the Arsenal boss is also concerned that the midfielder may miss out on Tuesday's home Champions League tie with Belgian club Anderlecht.
The 22-year-old has a niggling knee problem and Wenger said: "There is a slight problem with Wilshere. I don't think he'll be available against Burnley.
"It's very, very short term. He took a kick on his knee. I don't know if Tuesday will be too early or not. He's not training at the moment."
Wenger stressed that Theo Walcott will definitely not start against Burnley but hinted that he may get some game time as a substitute.
The England striker has not played since early January and remained on the bench for the entire 90 minutes of last week's trip to Sunderland.
Wenger added: "He's been out for too long to start the game. If I can give him some competition I'm happy to do it.
"The patience of the player is tested because they feel ready and want to play but usually realise much later that they were not completely ready.
"I know Theo well, he's a very intelligent player and he knows he's not completely there yet."
As for the club's form in the Premier League this campaign, the Frenchman believes the Gunners need to improve on their overall consistency before even thinking about their title chances this season.
Arsenal are fifth in the table, nine points adrift of leaders Chelsea, and Wenger went on: "We have to get consistent results to get the chance to come back.
"I'm focused on the quality of our games and consistency of our results before we even think about catching Chelsea.
"I don't write us off, let other people do that, but Chelsea are in a strong position at the moment. I believe it would not be realistic to just think now that we'll catch Chelsea very quickly."
Wenger is preparing his team to face a fired-up Burnley side still searching for their first Premier League victory.
"It's absolutely normal that other teams try to stop us from playing our game. I can understand that completely," Wenger said.
"We want to be good enough to get around that and not allow other teams to stop us playing.
"We are good enough, but we know the quality of our passing and our decisions must be better."
