The Springbok flanker dislocated his thumb and damaged ligaments in his right hand at the weekend.
Burger sustained the injury in the Stormers' Super 15 semi-final defeat to the Crusaders on Saturday and is now a serious doubt for the start of South Africa's Tri Nations campaign in less than three weeks time.
His club side confirmed the injury but are yet to comment on how long they expect Burger to be out of action for.
The Boks begin the 2011 Tri Nations against Australia in Sydney on July 23 and it would be a surprise if Burger was rushed back for that fixture with the World Cup now under two months away.
South Africa then play New Zealand in Wellington a week after the Tri Nations opener before hosting the Wallabies in Durban on August 13 and the All Blacks in Port Elizabeth on August 20.
The World Champions then start the defence of their global crown against Wales in Wellington on September 11.
Burger has been an instrumental figure for his country since making his debut in 2003. The International Rugby Board's World Player of the Year in 2004, Burger has won a total of 74 caps and was a World Cup winner in 2007.
Despite being injured in the April prior to the 2009 Lions tour of South Africa, Burger started the second Test but was yellow carded inside the opening few minutes in Pretoria and was subsequently found guilty of making contact with the eye area of Lions wing Luke Fitzgerald.