Vušković's great-grandfather Marko made three appearances for Hajduk Split on the liberated territory of Vis during World War II, and later worked as a club executive. Vušković's grandfather Mario, whom he was named after, attended the Hajduk academy in the 1970s and won a junior league title during Tomislav Ivić's tenure. Vušković's father Danijel is a former player who played as a defender. His cousin Moreno plays as a forward at Inter Zaprešić.