Wilshere and Podolski to miss Bayern trip
"He has an inflamed ankle, the other ankle to the one he injured before," Wenger told reporters before his squad left for Wednesday's last 16 second leg in Germany.
"He will be out for three weeks and will not be available against Munich, Swansea and not for England."
England play San Marino on March 22 and Montenegro four days later.
Arsenal are optimistic that Wilshere, who is currently having treatment in Dubai, will be fit to return for their Premier League match against Reading on March 30.
Wilshere damaged his left ankle during Arsenal's 2-1 defeat at arch-rivals Tottenham Hotspur in the North London derby on March 3 and has been slow to recover.
An injury to his right ankle sidelined him for the whole of last season.
Wenger added: "He has some bone bruising. The last time it deteriorated into a stress fracture. That's what we don't want to happen.
"We trust people who are specialists in this area. He was out for 17 months and we had to deal with it and so we will deal with it for two or three games."
Striker Lukas Podolski, who has been relegated to the bench for the last six matches, is also suffering with an ankle problem and is a doubt as Arsenal attempt to overcome a 3-1 first leg home defeat in Munich.
Arsenal will also be without defender Bacary Sagna for the match as he is suffering with knee and leg problems.
Wenger said Arsenal would have to make the "impossible possible" to stay in the competition.
"The chance is small," said Wenger, "but the chance makes the impossible possible. That would be a great credit for our team if we do it. We have to try to achieve it."