Pilkington suffered a tear in their 2-1 defeat against Newcastle United at the weekend.
Norwich boss Chris Hughton does not expect the midfielder to be back in contention until early in the new year.
Hughton told the club’s official website: “Anthony has had a scan which shows a grade two strain which usually means five or six weeks out.”
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Summer 2016 is set to be a very exciting time for football fans, as this is when the long awaited Euro 2016 tournament kicks off.
Since Guus Hiddink arrived at the helm, Chelsea have improved considerably after a tricky start to the campaign.