His club said tests had revealed “significant cartilage damage” but it is not the same injury which saw him miss seven months of this season.
Welbeck said on social media: “I will be back stronger!”
Arsenal manager Arsene Wenger had said he was “worried” about Welbeck after Sunday’s game at Etihad Stadium.
The former Manchester United player returned to action in February having been out since May 2015 with an injury to his left knee.
He had been expected to make Roy Hodgson’s 23-man squad for the European Championship in France after scoring five goals in 16 games for the Gunners this term.
Welbeck was England’s second-top goalscorer in Euro 2016 qualifying with six goals in five matches.
England will also be without Welbeck’s Arsenal team-mate Alex Oxlade-Chamberlain after the midfielder suffered his second knee injury of the season last week.