Welbeck went off in Sunday's 2-2 draw at Manchester City after sustaining "significant cartilage damage", the London club said on their website (www.arsenal.com) on Wednesday.
Since returning in February after a knee injury suffered in May last year, Welbeck has been in good form, scoring five times including winners against Leicester City and Norwich City.
Welbeck was England's second-top scorer in Euro 2016 qualifying, with six goals in five matches, and would almost certainly have been part of Hodgson's 23-man squad for the European championship which gets underway in June.
The England boss, who will name his final selection on Monday, now has a decision to make about Welbeck's replacement.
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With captain Wayne Rooney, Tottenham Hotspur's Harry Kane, Leicester City's Jamie Vardy and Liverpool's Daniel Sturridge likely to be named in the squad, the England manager must decide whether to replace Welbeck with another central striker.
While Welbeck was comfortable playing through the middle, he has frequently been utilised out wide, which could tempt the England manager to include a player like Arsenal's Theo Walcott, who would also cover both positions.
This might rule out players such as West Ham United's Andy Carroll or Sunderland's Jermain Defoe, who are both deployed as out-and-out centre forwards.
England face Russia, Slovakia and Wales in Group B of the 24-team tournament in France that starts on June 10.
Third-placed Arsenal end their league campaign at home to relegated Aston Villa on Sunday.
(Reporting by Shravanth Vijayakumar in Bengaluru/ Toby Davis in London; Editing by Catherine Evans)
