Kostunica's proposal that talks be held in Kosovo and Serbia comes after almost 18 months of mediated discussions, held mostly in Vienna but also in Brussels and London, failed to reach an accord.
On Friday, a report made available at the UN said there was no compromise to be found between Serbia's offer of autonomy and the Albanian demand for independence after talks approached the December 10 deadline.
The United States and almost all EU member states support independence for the territory, seeing it as the best way to ensure stability in the Balkans.
Kosovo Albanians say they will declare independence within weeks.
But Serbian-ally Russia renewed requests on Friday for talks to continue until the two sides reach an agreement.
Russia said it wants any final solution for Kosovo to go through the UN security council, where it has a veto.
"It is of the greatest importance that the security council support the proposal of Russia," Kostunica said.
Serbia gives "its full support to this constructive Russian initiative which aims to avoid a great crisis."
In Kosovo, a 550-strong German reserve force, temporarily reinforcing 16,000 Nato peacekeepers, carried out helicopter and ground exercises centred on Prekaz - once the heartland of the Kosovo Albanian guerrilla resistance to Serb rule.
"Unfortunately, the fixation of some Western capitals - above all Washington - with independence for Kosovo and that there is no alternative to such independence ... is now the main hindrance on the path to a negotiated settlement"
Sergei Lavrov, Russian foreign minister
Nato ministers on Friday pledged to keep the peace force at current strength, and make more troops available to deal with any outbreak of violence.
Analysts dismiss talk of war, but say low-level violence is to be expected.
On Saturday, Sergei Lavrov, the Russian foreign minister, said the United States was hindering the search for a solution to the ongoing conflict.
"Unfortunately, the fixation of some Western capitals - above all Washington - with independence for Kosovo and that there is no alternative to such independence ... is now the main hindrance on the path to a negotiated settlement," he said.
The UN security council is due to discuss the mediators' report on December 19.
But Washington and the major EU countries have indicated they are ready to recognise Kosovo without a new UN resolution, and the EU is preparing to deploy a mission to take over supervision of the territory from the UN.
Diplomats say Kosovo could declare independence in late January at the earliest.