The 25-year-old Denmark international has not played since September 25 when he came on as a substitute in the 2-2 draw at home to Sunderland.
He visited a specialist in London on Wednesday to assess the problem and has been told not to train to allow the problem to rectify itself.
"Agger is long-term. They can't put a date on it," said Hodgson.
"At the moment he can't train, he has just got to rest and wait for the next time he goes to see the specialist again.
"They are very anxious that bone doesn't form. It is a calcification issue and training would certainly accelerate the calcification.
"Now they have to wait and hope the body breaks down the calcifying elements before he can even think about training again.
"So it could be very long term."