Stošić was born in Vranje, Socialist Federal Republic of Yugoslavia. During his early career he represented Red Star Belgrade, who also loaned him to FK Rad, FK Radnicki Nis and a club in Australia. In Red Star's 1990–91 victorious campaign in the European Cup, he played the last minutes of the final match, which went into extra time as the team emerged victorious over Olympique de Marseille after a penalty shootout.