The U.S. ambassador to the UN, Zalmay Khalilzad, said he believes a majority of the Security Council's 15 member nations will support the proposal for internationally supervised independence for Kosovo.
But he said it remains unclear whether Russia will abstain or veto the measure.
Russia's UN Ambassador Vitaly Churkin declined to say whether Russia would consider vetoing the proposal. Moscow has said it supports further talks between Serbs and Kosovo's ethnic-Albanian majority.
The ethnic-Albanian majority has welcomed the proposal from a UN mediator that the province be put on the road to independence. But the proposal is strongly opposed by Serbians, who have close historical and cultural ties to Kosovo.
U.S. Undersecretary of State Nicholas Burns said in Zagreb today that Russia should be "constructive" when the UN Security Council votes on the province's future.
Kosovo has been administered by the UN and NATO since NATO's 1999 air campaign ended a Serbian crackdown on ethnic Albanian separatists.