"He has had no setbacks and is getting better. He will be available after West Brom," Wenger told a news conference on Wednesday.
"Overall the progression now should be normal. He's available for selection -- I don't think for tomorrow -- but from the next game on it's a question of do I select him or not?"
Wenger said Wilshire would be fully fit for the June 10-July 10 European Championship in France, and England manager Roy Hodgson would not be taking a risk if he included the player in the squad.
England face Russia, Slovakia and Wales in Group B of the 24-team tournament and Hodgson will name his squad on May 12.
"The European Championship starts on June 10 and there's still nearly two months," Wenger added. "If you look at his fitness at the moment I think in two months he should be perfectly fit. If he has no setbacks it would not be a gamble."
Wenger confirmed that Arsenal were in contract talks with playmaker Mesut Ozil and forward Alexis Sanchez and that midfielder Santi Cazorla, who has not played since the end of November, had returned to training.
The Frenchman hoped there was still a late twist in the battle for the league title left, with Arsenal trailing leaders Leicester City by 13 points, with both teams left with four games to play after the Gunners face West Brom.
"The season is far from over," Wenger said. "It depends on who you play and the commitment you face."
Wenger took responsibility for Arsenal's last two league games against West Ham United and Crystal Palace, which ended in draws that all but ended their title hopes.
"Against West Ham and Crystal Palace we were not efficient enough. That is a regret. It's my job to get the team 100 percent ready," he added.
"Every time it doesn't work you question yourself, it is part of the job. In the end it is always down to me and I stand up to that."
(Reporting by Simon Jennings in Bengaluru)
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