Wilshere scored twice as the Gunners beat Marseille 2-0 on Tuesday, and Wenger feels that the England international and his Welsh teammate are in competition with each other this season.
"[Wilshere's] starting to think 'what Ramsey can do I can do as well', he's calm in front of goal and the first goal shows that," Wenger told reporters.
"It's a joke about Aaron but it puts in his mind that he has to score goals as well. Certainly, when you're a midfielder you want to score goals.
"No [I don't think Wilshere will be a goalscorer] but I have nothing against it. If I'm wrong then I'm happy."
A point for the Gunners in their next Champions League game away to Napoli will see them secure qualification.