Kostunica is accompanied by several of his ministers on the trip, which was announced as the first official visit since his country became independent by default when Montenegro declared independence from the union last weekend.
Immediately after the Montenegrin referendum, Republika Srpska Prime Minister Milorad Dodik launched the idea of a similar referendum in the Bosnian Serb ministate.
Independence for the half of Bosnia currently run by ethnic Serbs would create the precondition for it to join up with Serbia - an idea that prompted the brutal 1992-95 Bosnian war, the AP reports.
Right-wing Bosnian Serb extremists immediately organized a petition on the streets of Banja Luka, demanding a referendum. Thousands signed.