Kiev has said an "anti-terrorist operation" against pro-Moscow separatists is under way, as Russia declared Ukraine on the brink of a civil war.

The Kiev crackdown appeared to get off to a slow start, if at all, however.

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

Oleksander Turchinov, Ukraine's interim president, insisted the operation had started in the eastern Donetsk region, although he said it would happen in stages and "in a considered way".

Amid the deepest crisis between Russia and western countries 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 Slovyansk, 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 in a statement posted on his Facebook page on Tuesday.

Ukraine has accused Russia of stirring up the separatists following its annexation of Crimea, while Moscow says Kiev has provoked the crisis by ignoring the interests of its citizens who use Russian as their first language.

Pro-Russian politicians assaulted

Elsewhere in Kiev, two pro-Russian politicians have been attacked by pro-Ukrainian activists, the Associated Press news agency reported.

Oleh Tsaryov, a pro-Russian politician and a candidate in the May 25 presidential elections, was beaten by dozens of enraged activists in the early hours on Tuesday as he was leaving a television studio. The activists pelted him with eggs, shouted insults and then assaulted him.

Tsaryov's press service said in a statement that he was "brutally beaten".

Another Russian-leaning politician and presidential hopeful, Mikhaylo Dobkin, was sprayed with a green disinfectant and had flour thrown at him late on Monday.

Source: Agencies