Dunne suffered the injury in Sunday's home clash with Premier League leaders Manchester City after falling heavily when challenging England keeper Joe Hart for a cross and will undergo surgery in Birmingham on Monday.
A club statement confirmed: "Richard Dunne sustained a fracture to his right clavicle in the game against Manchester City. He will undergo an operation today in Birmingham. It is anticipated that he will be sidelined for up to eight weeks."
Meanwhile, midfielder Chris Herd has returned to full training having recovered from an ankle ligament injury which has left him sidelined since December.
He will be available for the game against Wigan on February 25.