 Džumhur was born in Konjic, Bosnia and Herzegovina.When he was only two months old his father, Imam Abduselam Džumhur (1885-1933) and mother Vasvija ("née" Tufo) (1900-1978) moved to the capital of Yugoslavia, Belgrade, where his father got a job in the Yugoslav Army.Zuko Džumhur finished elementary school and the first four grades of high school in Belgrade, then moved to Sarajevo where he finished high school in 1939.