"It's not looking good. He will see a surgeon on Monday with the possibility of doing something on Tuesday," boss Paul Lambert told the club website.
Lambert added he would be looking for cover in the loan market, but does not expect movement before the weekend.
Barnett has been a first choice centre-half for the City manager since he joined on loan from West Brom in August.
At this stage of the season we'll look to bring somebody in with experience
Norwich boss Paul Lambert
The former Luton man made his switch to Carrow Road permanent in January and has now played 26 games for the Canaries in total.
"It's a huge blow for him because of the way he's been playing for us and for myself and the club. But that's football," Lambert added.
"We're trying [to bring someone in] at the minute and hopefully something will transpire over the next few days, and hopefully we'll get somebody in.
"At this stage of the season we'll look to bring somebody in with experience."
Meanwhile Lambert confirmed midfielder Matt Gill has returned from his loan spell at League One Walsall.
Source: BBC Sport
Source: BBC Sport