The 25-year-old sustained a hamstring injury in the 2-1 win over Reading on Saturday.
“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.
Barnett joined Norwich on loan from West Brom in August.
“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.”