KIEV (REUTERS) - Russia declared Ukraine on the brink of civil war on Tuesday as Kiev said an "anti-terrorist operation" against pro-Moscow separatists was underway, though the crackdown got off to a slow start, if at all.

Twenty-four hours after an Ukrainian ultimatum expired for the separatists to lay down their arms, witnesses reported no signs yet that Kiev forces were ready to storm state buildings in the Russian-speaking east that the rebels have occupied.

Police said separatists had voluntarily surrendered at the police headquarters in the city of Kramatorsk. Interim president Oleksander Turchinov insisted the operation had started in the eastern Donetsk region, although it would happen in stages and "in a considered way".

Amidst the deepest East-West crisis since the Cold War, the leaders of Russia and the United States called on each other to do all in their power to avoid further bloodshed.

Russian Prime Minister Dmitry Medvedev gave a gloomy assessment after at least two people died on Sunday when Kiev unsuccessfully tried to regain control in Slaviansk, one of about 10 towns and cities where the separatists have seized buildings.

"Blood has once again been spilt in Ukraine. The country is on the brink of civil war," he said on his Facebook page.

Mr Turchinov said the offensive, which he first announced on Sunday, was finally underway. "The anti-terrorist operation began during the night in the north of Donetsk region. But it will take place in stages, responsibly, in a considered way. I once again stress: the aim of these operations is to defend the citizens of Ukraine," he told parliament.

On Tuesday morning, a Reuters correspondent in Slaviansk had heard no shots or explosions in the town, which lies about 150 km from the Russian border. Outside the occupied local police headquarters, about a dozen civilians manned barricades that have been built up overnight with more tyres and wooden crates.

A dozen or so armed Cossacks - paramilitary fighters descended from Tsarist-era patrolmen - stood guard at the mayor's offices. Shops were functioning as usual and bread supplies were normal.

Ukraine's security forces have been in some disarray since protesters ousted pro-Moscow President Viktor Yanukovich in February. However, the delays to the crackdown may also reflect a desire by the interim leadership to avoid making things worse by causing civilian casualties.

Moscow accuses Kiev of provoking the crisis by ignoring the rights of citizens who use Russian as their first language, and has promised to protect them from attack. It has also highlighted the presence of far-right nationalists among Kiev's new rulers.

However, a United Nations report on Tuesday cast doubt on whether Russian-speakers were seriously threatened, including those in Crimea who voted to join Russia after Moscow forces had already seized control of the Black Sea peninsula.

"Although there were some attacks against the ethnic Russian community, these were neither systematic nor widespread," said the report by the UN human rights office. The report, issued after two visits to Ukraine last month by Assistant Secretary-General for Human Rights Ivan Simonovic, cited "misinformed reports" and "greatly exaggerated stories of harassment of ethnic Russians by Ukrainian nationalist extremists".

These, it said, "had been systematically used to create a climate of fear and insecurity that reflected on support to integration of Crimea into the Russian Federation".

Russian Foreign Minister Sergei Lavrov denied accusations from both Kiev and the West that Russia was stirring up the separatists in the east and south-east as a possible prelude to repeating its annexation of Crimea.

"Ukraine is spreading lies that Russia is behind the actions in the southeast. This is a total lie that supposes that those residents there are completely incapable of protesting of their own will," Mr Lavrov said on a visit to China. He called on Kiev to hold back before a meeting between Russia, the European Union, the United States and Ukraine planned for Geneva on Thursday.

"You can't send in tanks and at the same time hold talks," he said. "The use of force would sabotage the opportunity offered by the four-party negotiations in Geneva."

Moscow says it wants constitutional change in Ukraine to give more powers to Russian-speaking areas, where most of the country's heavy industry lies, while the secessionists have demanded Crimean-style referendums in their regions.

Kiev opposes anything that might lead to the dismemberment of the country. But in an attempt to undercut the rebels' demands, Mr Turchinov has held out the prospect of a nationwide referendum on the future shape of the Ukrainian state.

Mr Lavrov said Kiev's apparent willingness to "resolve through negotiations all the problems relating to the legal demands of the inhabitants of the south-east regions of Ukraine, is certainly a step in the right direction, albeit very belated".

US President Barack Obama criticised Russia in a telephone call with President Vladimir Putin on Monday, saying its actions were not conducive to a diplomatic solution.

"The president emphasised that all irregular forces in the country need to lay down their arms, and he urged President Putin to use his influence with these armed, pro-Russian groups to convince them to depart the buildings they have seized," the White House said in a statement.

Moscow put the onus for peace on Washington.

"President Putin called on Barack Obama to do his utmost to use the opportunities that the United States has to prevent the use of force, and bloodshed," the Kremlin said in a statement.

Nato states have sent troops, aircraft and ships to eastern Europe to reassure nervous post-communist alliance members, including the Baltic states - which were once Soviet republics - as well as Romania and Bulgaria. A Russian fighter aircraft made repeated low-altitude, close-range passes near a US ship in the Black Sea over the weekend, the Pentagon said, condemning the action at a time of heightened US-Russian tensions. The White House has warned Russia it would face further costs over its actions in Ukraine, but it made clear that the United States was not considering lethal aid for Ukraine.