He said he will talk with all parties in parliament -- including the Serbian Radical Party and the Socialist Party, whose leaders -- Vojislav Seselj and Slobodan Milosevic, respectively -- are both in detention in The Hague on charges of war crimes.
"The DSS today began consultations on forming a government. I want to add that we will hold talks with representatives of all parliamentary parties, with no exceptions," Kostunica said. "We will not make public comments on the contents of these talks until we reach an agreement and until we have talked with everyone."
Kostunica's moderately nationalist DSS came in second in the early parliamentary election on 28 December.
The ultranationalist Serbian Radical Party won nearly one-third of all seats in parliament but is unable to form a government, even with the support of their former allies, Milosevic's Socialists.
Kostunica said the priority of a national unity government would be to draw up Serbia's new constitution to ensure political stability. The current constitution dates from Milosevic's authoritarian rule.
"A broad-based government is the way to solve the constitutional issue in Serbia," Kostunica said. "For a [new] constitution to be adopted, more than a simple parliamentary majority is needed, more support is needed than any majority or minority government could possibly have. We need more. We need a consensus if we want the constitution to last. We need it for stability, because the constitution means stability, and that is why I mentioned the possibility of forming a broad-based government."
Most parties, including the Radicals, immediately rejected the idea. The Democratic Party -- the main party of the outgoing pro-reform cabinet -- yesterday said the plan was unacceptable, as it would bring back to power Milosevic's Socialists and their Radical Party allies, who dragged Serbia into the wars of the early 1990s.
If they joined forces, the four main pro-reform parties could form a majority government -- but chances for that are, at least for now, rather slim.
The former allies, who together toppled Milosevic, are now split by ideological differences and personal antagonisms. It was their continuous infighting that, in part, drove disillusioned voters to back the Radicals.
Kostunica warned that if the talks reach a dead end, another general election may have to be called -- but said that would not be a good solution.
Liljana Bacevic, the head of the Center for Public Opinion Research in Belgrade, told RFE/RL that Kostunica may have floated the idea of a national unity government as a way to avoid a new election.
She said currently it is the DSS and the liberal G-17 Plus who are particularly eager to form a government, as they are both expected to lose votes in a new election.
"For Kostunica and G-17 Plus, the best solution is to form any government with the seats in parliament that they have now. In a future parliament after the next election, they will have less votes and they will have less seats and that is why, I suppose, they would like to form any government which will be a better solution for them than to run in a [new] election," Bacevic said.
G-17 leaders were the first to meet with Kostunica for coalition talks earlier this week. Local media reported that the two parties had even managed to strike a deal, with the DSS giving G-17 Plus the post of prime minister, if in exchange they could nominate their candidate as speaker of the new parliament and current acting president of Serbia.
Striking a deal with the other two pro-reform parties would be more complicated. Vuk Draskovic, the leader of the Serbian Renewal Movement, has said a coalition government should include all four democratic parties, who would share responsibilities according to their seats in parliament.
The Democratic Party, Kostunica's bitter rival, previously said it would not join a coalition but would rather support a minority cabinet -- if it pursued the pro-reform policies of the outgoing government. Kostunica himself, in the run-up to the election, ruled out a coalition with the Democratic Party.
Late yesterday, however, the leaders of the two parties met for their first official discussions.
Worried by the Radicals' election gains, the United States and the European Union have urged democratic parties to overcome their differences. Analysts expect that, under international pressure, they will end up reaching some sort of an agreement.
The question remains, however, how long such a fractured coalition could survive. Many fear a new election could further strengthen support for the Radicals.
Bacevic said not just the Radicals and their allies, but also the democratic bloc parties, would emerge strengthened and more clearly defined from a new election.
"In a future election, so-called small parties which are mostly, if not exclusively, democratically oriented [but which] in this election did not pass the 5 percent of votes requirement will have enough time to negotiate, to organize themselves so as not to lose votes of democratically oriented voters. They could join either the Democratic Party or G-17 Plus, or even Kostunica's party, but not find themselves in the position that they are not represented in the parliament," Bacevic said.
Kostunica said the first round of coalition talks should be over by tomorrow, although political bargaining could drag on for weeks. Under current law, the new government should be in place by 1 March.