The Norwegian international damaged his ankle against Cardiff City late last month, but it was decided that surgery was needed and he went under the knife last week.
It will now be two to three months before the 27-year-old midfielder is back for Norwich.
"Alex (Tettey) got a blow on the ankle in a clash with Fraizer Campbell towards the end of the match against Cardiff," his advisor Jan Ludvigsen is quoted by VG.
"The doctors and the medical staff judged that a surgery was necessary and Alex just returned home after the operation in London on Thursday morning.
"That kind of injury usually requires eight to 12 weeks convalescence. Injuries are always saddening, particularly now that Alex had got a lot of confidence in Norwich and was playing really well.
"But he is a very focused person and will do everything he can to be back on the pitch as soon as possible."
Norwich boss Chris Hughton added: "Unfortunately for Alex we feared the worst immediately after he did the injury because he felt so sore for a few days afterwards.
"He has now had an operation on the area just above his ankle which will keep him out for perhaps a couple of months. Alex took a whack to an area just above his ankle and it is still very sore.
"He has damaged the ligament just above the ankle. I thought he was playing really well. We know the type of player Alex is and he was doing that role very, very well. It's a big blow for us and for the lad."