In the late 1980s, Savićević married Valentina "Vanja" Brajović. The couple had met and began dating a few years earlier in Titograd while Savićević played for FK Budućnost and teenage Vanja attended the local streamlined touristic high school. Their first child, son Vladimir, was born in November 1989 in Belgrade while Savićević played for Red Star. While living in Belgrade, Savićević and Brajović reportedly resided in an apartment they leased from Serbian professional handballer Svetlana Kitić who had been playing abroad in Italy during that time. Their second child, daughter Tamara, was born in 1992. The couple divorced in 2000.