Underhill, who has caught the eye of England head coach Eddie Jones, will leave a big hole in Ospreys' back row after he scored three tries in 16 games last season.
Ospreys already have a growing injury list with centre Owen Watkin also on the treatment table, and he could miss the whole campaign due to a cruciate ligament injury.
"Sam suffered a high grade AC joint injury in training last week, since when he has had scans and we've consulted with a specialist at Spire Cardiff," said Ospreys' medical performance manager Chris Towers.
"The agreed course of action for Sam is for him to undergo surgery to reconstruct the damaged ligaments. That will take place at the end of the week before he commences a programme of rehabilitation.
"The initial prognosis is that he can expect to be sidelined for a period of around four months."
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