Most Western states back the volatile area's drive for independence, and recently agreed that Kosovo's status would be determined by the European Union and NATO. Russia insists that Belgrade and Pristina continue to seek a compromise.
Mikhail Kamynin, Russia's Foreign Ministry spokesman, said that any replacement of the UN mission in Kosovo by the European Union could be discussed "provided an according agreement is reached between the sides."
The Security Council met late on Wednesday to review the work of the UN's interim mission in Kosovo, which has been deployed in the region since NATO's bombing of the former Yugoslavia ended a conflict between Albanian and Serb forces in 1999.
U.S. Ambassador to the United Nations Zalmay Khalilzad said earlier on Thursday that the Security Council was in a deadlock over Kosovo's future status, and no longer had any role to play on the issue.
This had also caused serious concern in Russia, Kamynin said.
Russian Ambassador Vitaly Churkin warned on Wednesday that Russia, which has veto power on the UN Security Council, would block any attempt by an independent Kosovo to become a member of the United Nations.
Meanwhile, Kosovo's newly elected Prime Minister Hashim Thaci said Pristina's independence was an accomplished fact and would be declared as soon as the United States and the European Union were ready to recognize it.