The Frenchman said that he agrees with the midfielder's remarks, adding that only an Englishman should manage the national team.
"I agree with him. I always said also an Englishman should manage the England team," he told BBC Sport.
Wenger went on to issue a warning that national teams could one day face a similar problem to Premier League clubs with a shortage of home-grown talent.
Wilshere sparked controversy when he said "if you live in England for five years, it doesn't make you English," with many linking the remarks to the news that Manchester United's Adnan Januzaj has been earmarked as a future England international.
However, the Arsenal man has since clarified his comments.
"Just to clarify a few things... again! Seems to be a trend in this country, poor journalism in my eyes! I wasn't referring to Janujaz," he wrote on Twitter.
"The question was should foreign players be allowed to play for England, and in my opinion I don't think they should! He is a great player...
"I wish he was English! I'm sure most of you agree with me even if some journalist are trying to make me look bad... again!"
Wilshere has 10 caps for England at senior level.