The injury also rules the 21-year-old out of Arsenal’s Champions League tie against Bayern Munich on Wednesday.
The Premier League side trail the Germans 3-1 from the first leg.
Wilshere picked up the problem in the Gunners’ 2-1 defeat by Tottenham on 3 March.
Despite having only seven caps to date, Wilshere is seen as a key component of Roy Hodgson’s midfield as England try to win a place in the 2014 World Cup finals.
They face San Marino on 22 March and then travel to Montenegro four days later for a game which could decide who tops Group H.
This latest blow is a yet another set-back for Wilshere in a career that has been hampered by injury problems.
He made his debut for Arsenal as a 16-year-old at Blackburn Rovers in September 2008 before being loaned to Bolton in January 2010.
On his return to the Emirates in August, Wilshere broke into the Arsenal team and made 49 appearances in all competitions that season, before being named PFA Young Player of the Year in 2011.
But he missed all of last season with a stress fracture of his ankle and had not recovered in time to represent England at Euro 2012 in the summer.