Wilshere has been linked with a move to newly-promoted Wolves this summer, as speculation over his future continues to grow.
The England international revealed that Wenger told him he could leave at the start of the season, but the club have since offered the 26-year-old a new long-term deal.
Wenger expects a decision to be made before the World Cup, but says he has not given Wilshere a deadline.
“I cannot give you any news because I don’t know what is in his mind,” Wenger said.
“Until now he has not signed but we want him to stay. He has an offer for a long-term contract and I hope at some stage he will sign it.”
When asked if he was confident Wilshere would sign a new deal, Wenger said: “I am always confident. As a club we want the players to stay and to be paid well and I believe that everybody would agree that we pay our players very well.
“He has the freedom to make that decision. I want him to stay.
“He is focused on the next game and will certainly make his decision later.
“I think he will make it before the end of the season. He won’t want to go to the World Cup and not have settled his situation. That would not be reasonable.”
The Arsenal boss also faced more questions over his own future at the club, but insists he is only focused on finishing the season well.
“My personal situation is not so much my worry at the moment,” Wenger said. “My worry is to transform the season into a success.
“My future is in front of me every time I make a half turn. That is not my biggest priority.
“I am responsible for the results and when they are not as good as everybody wants, people ask that question.
“I cannot master that. I can only master my commitment, the quality of my work and I do that.”