Wilshere was hurt when we went into a strong tackle with Danish defender Daniel Agger in the first half but after treatment he continued playing until the 58th minute.
"It is a blow for him and for us and we just want him to recover as quickly as possible to prepare for the rest of the season.
"You can only be very down in a situation like this, especially in the first days. He has gone away for a few days to get over the disappointment and to think about something else."
Wilshere, who will be out of action for up to eight weeks, has suffered a numbrt of injuries in his career including several to his ankle and foot.
"This was an accident on his good ankle," Wenger said. "The tackle was strong, but he (Agger) went for the ball."
Wenger also said he had no problem with the way the England physios handled the situation on the pitch.
"It's the player that gives you the indication of whether he's in pain and you trust the player to tell you how he feels after a few minutes," he said.
"We are not at the point where you can have an instant checkup with an X-ray to see how big the damage is. But most of the time the player indicates if he can go on or not."
The injury could hamper Wilshere's preparations for the World Cup, but he should be fit if coach Roy Hodgson picks him for Brazil.
A number of England players have been injured in build-ups to the World Cup since 2002, including David Beckham and Wayne Rooney who recovered from broken metatarsals to play in 2002 and 2006 respectively.
Rooney also damaged his ankle ligaments before the 2010 World Cup but was fit for the tournament.
Wilshere's injury comes at a bad time for Arsenal, who are third in the Premier League and four points behind leaders Chelsea with 10 games remaining.
He will also miss the Champions League last-16 clash against Bayern Munich and the North London derby against Tottenham Hotspur as well as matches against Chelsea, Everton, Manchester City and West Ham United.
Arsenal defender Laurent Koscielny is doubtful for the Everton match after returning from international duty with France with a hamstring strain but midfielder Aaron Ramsey, out since the end of December, should be fit to return next weekend.
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- Daniel Agger