The 21-year-old picked up the knock during the north London derby against Tottenham Hotspur 10 days ago and has since been in Dubai receiving treatment.
He is also a doubt for the weekend Premier League game against Swansea City, and as such will likely be left out of Roy Hodgson's England squad to play a World Cup-qualifying header the week after.
Wilshere's absence will just about end any hope Arsene Wenger's side have of overturning a 3-1 Champions League last 16 first-leg deficit against the Bundesliga leaders.
Only a handful of sides have ever come through after conceding such an advantage in the first meeting, and Wenger would not jeopardize the long-term fitness of Wilshere in what looks like a futile task.
Instead, it looks as though the Gunners' academy graduate will be given until the end of the month to recuperate and will be identifying the March 30 home game against Reading as one potential comeback date.
The Spurs' defeat to Liverpool on Sunday has left the door open for Arsenal to still rescue a top-four spot, and that achievement seems vastly more attainable than going to the Aliianz Arena and scoring three without reply.
Arsenal currently sit fifth in the league, seven points behind Spurs in third and five behind Chelsea in fourth. Both Arsenal and Chelsea have a game in hand over Andre Villas-Boas's side.
Kieran Gibbs, Bacary Sagna and Lukas Podolski are also touch and go to feature in Bavaria whilst Nacho Monreal is cup tied to give Wenger unwanted selection dilemmas ahead of what is bound to be one of his toughest assignments in 16 years at the Arsenal helm.
The Frenchman will undoubtedly stick to the party line that success against the Bundesliga champions elect is still attainable, but in reality this is a lost cause, and the Gunners will soon have to refocus their attentions on usurping either London rival to a top-four finish.
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