Ivan Ćurković joined his hometown FK Velež Mostar as a youth goalkeeper in 1958 and, two years later, was offered a professional contract. He remained with them through 1964, at which point he joined FK Partizan. This was also the year that he represented his country in the tournament at the 1964 Tokyo Olympics, having débuted with the Yugoslavian national team in 1963. At the Games, Yugoslavia was eliminated by upcoming bronze medalist Germany in the quarter-finals and ranked sixth overall. He was with Partizan through 1972, winning the Yugoslav First League in 1965.
Ćurković joined France’s AS Saint-Étienne in 1972, with his career with the Yugoslavian national team having ended in 1970 with 19 caps. It was in France that he had his greatest successes, winning the Ligue 1 four times (1974-76 and 1981) and the Coupe de France three times (1974-75, 1977) before retiring from active competition in 1981. He then embarked upon a career in sports administration, and was president of Partizan from 1989 through 2006. In 2005 he was appointed interim president of the Olympic Committee of Serbia and Montenegro, becoming full president in 2006. When Montenegro became independent in 2006, he remained the head of the Serbian Olympic Committee through early 2009. He was decreed a French citizen in 1982 and received the Legion of Honour in 2005.