The 25-year-old has only made 11 Premier League appearances for the Gunners this season having only recently return from another major knee injury sustained in April 2014.
It will come as a blow to the England set-up as the former Manchester United striker has been an important part of Hodgson's squad when available.
Welbeck scored six goals during the qualifying campaign for Euro 2016 and was expected to be the fifth striker in the squad alongside Harry Kane, Wayne Rooney, Jamie Vardy and Daniel Sturridge.
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