Oxlade-Chamberlain, 18, was handed his first Premier League start for the Gunners last week against Manchester United and put in an outstanding performance, though his side were beaten 2-1.
Wenger has bided his time with the Oxlade-Chamberlain since signing him from Southampton for £12 million last summer, using the midfielder sparingly, but has intimated that he may now look to blood him more regularly.
And should he start to consistently star for Arsenal, Wenger believes there is no reason why the England Under-21 international can't stake a claim for a place in Fabio Capello's squad in Poland and Ukraine.
"You cannot rule it out and it is not because you are young that you cannot play. We will see in the next three months and if he does well, why not?'' Wenger said.
"We live in a world of immediacy. I would not have expected him to be so mature in September because he still looked like a boy. Now suddenly he looks like he can take pressure and that he has the personality to produce under pressure.
"But he has only really played one game at the top, so let's see how well he does.''
Wenger had long been criticised for packing his side with overseas talent, but now the likes of Jack Wilshere, Walcott, Kieran Gibbs, Carl Jenkinson and Oxlade-Chamberlain all, when fit, compete for regular first-team action.
"They get a chance here that is for sure,'' Wenger said. "Young players want to join us because they know they will get a chance at the top level.
"I believe for an English national team player, you have to start to play at 18 at the top level if you want to become an international player. They all played at 18 - John Terry, Rio Ferdinand, Wayne Rooney, Steven Gerrard.''
"I believe that when Oxlade-Chamberlain was making his choices, he knew that he would have a chance here. I think we have always shown we are willing to take out experienced players and put younger ones in when they deserve it.''