Earlier, Avakov had warned residents to stay indoors. "Pass this on to all civilians: they should leave the center of town, not come out of their apartments, and not go near the windows," he wrote on his Facebook page.
In Slovyansk Sunday, few appeared to heed the warning — as families could be seen walking around and shops were open. Barricades could be seen around the city on roads leading toward the city center, manned by pro-Russian protesters.
In the nearby town of Kramatorsk, militants exchanged gunfire with police late on Saturday, though there was no confirmation anyone had been hurt.
Kyiv accuses Moscow of trying to deepen violence and chaos in Ukraine, a former Soviet republic it once ruled. The Kremlin, it says, wants to undermine the legitimacy of presidential elections on May 25 which aim to set the country back onto a normal path after months of turmoil.
Russia's foreign ministry called the planned military operation a "criminal order" and said the West should bring its allies in Ukraine's government under control.
"It is now the West's responsibility to prevent civil war in Ukraine," the ministry said in a statement. Russia moved to put an urgent discussion of the "extremely dangerous" situation on the agenda of the United Nations Security Council. The council met in New York on Sunday to discuss the crises.
Earlier, the American ambassador to the U.N., Samantha Power, said on ABC's "This Week" that the latest events in Ukraine bore "the telltale signs of Moscow's involvement."
"The president has made clear that, depending on Russian behavior, sectoral sanctions in energy, banking, mining could be on the table, and there's a lot in between," she added.
Relations between Russia and the West are at their worst since the end of the Cold War, the result of the crisis that began when the Moscow-backed Ukrainian president was pushed out and Russia annexed Crimea from Ukraine.
Moscow justified sending its military into Crimea, on Ukraine's southern tip, by saying the Russian population there were under threat, and some in Western governments believe the Kremlin is preparing a similar scenario for eastern Ukraine.
In Washington, the White House expressed concern that the seizures of public buildings in eastern Ukraine could be a prelude to a Russian military incursion, though Moscow has strenuously denied any such intention.
NATO says Russian armed forces are massing on Ukraine's eastern border, while Moscow says they are on normal maneuvers.
On Saturday, in what had the look of a coordinated action, pro-Russian militants, some of them armed, seized public buildings in several towns, challenging Kyiv's already shaky control over the region.
In Kramatorsk, about 7 miles south of Slovyansk, gunmen seized the police headquarters after a shootout with police, a Reuters witness said.
The attackers were a well-organized unit of over 20 men, wearing matching military fatigues and carrying automatic weapons, who had arrived by bus. Video footage showed the men taking orders from a commander. Their identity was unclear.
Their level of discipline and equipment was in contrast to the groups who have occupied buildings so far in Ukraine. They have been mostly civilians formed into informal militias with mismatched uniforms.
Ukrainian police appeared powerless to intervene, and it was not clear if local security forces were still taking orders from Kyiv. The police chief for the Donetsk region, which includes Slovyansk, resigned after pro-Russian protesters picketed his headquarters.
In Slovyansk, militants blocking the road leading into the city from the north set fire to tires and trees, hoping the smoke would hamper any operation by government forces.
"We want to join Russia. We would be very grateful if Russia helps us," said a gunman in the center of Slovyansk who gave his name as Alexander, wearing camouflage fatigues and a black balaclava mask. "We will stand until victory. Honestly, it's not scary for me to die for freedom."
Ukrainian commentator Sergei Leshchenko said the burst of activity by pro-Russian groups was an attempt by the Kremlin to secure a strong negotiating position before international talks about Ukraine. Representatives of Russia, Ukraine, the United States and the European Union are due to meet in Geneva on Thursday for the talks.
"Russia will come to the talks with the position that 'Donetsk and several neighboring regions are already ours — now let's talk about federalization'," said Leshchenko, a commentator with the Ukrainska Pravda newspaper.
Al Jazeera and wire services