The Ireland international has made nine appearances for Chris Hughton's team this season - scoring one goal - but will now be absent until the new year.
Norwich boss Hughton was able to offer more positive news on Robert Snodgrass, another injury victim.
A knee injury had been bothering the Scotland international and he did not feature against Newcastle, but Hughton expects Snodgrass to be back in mid-December.
"Anthony has had a scan which shows a grade two strain, which usually means five or six weeks out," the manager told Norwich's official website.
"Robert came back from international duty with Scotland and felt his knee at the beginning of training on Friday.
"He travelled with us to Newcastle as we hoped it would settle down but unfortunately it didn't.
"He could be out for a few weeks, but not as long as Anthony."
The loss of the midfield duo comes at a bad time for Norwich, with matches against fellow strugglers Crystal Palace and Sunderland among the club's upcoming fixtures.
