Ukrainian troops have surrounded the pro-Russian stronghold of Sloviansk in the east, with residents saying they fear they will be stormed.
At present the troops appear to be concentrating on retaking government buildings from separatists in smaller towns around Sloviansk.
Pro-Russian groups have reacted angrily to the deaths of many separatists in a fire in Odessa in the south on Friday.
Dozens were killed after barricading themselves in a building in the city.
The BBC's Sarah Rainsford in the regional capital Donetsk says that while it appears Ukrainian forces have sealed off the roads in and out of Sloviansk, they are moving around the city and concentrating on smaller towns nearby.
Our correspondent spoke to a resident of Sloviansk who said people there were expecting the town to be stormed.
Several people were reported killed in clashes around Sloviansk on Saturday. The defence ministry said one checkpoint was removed overnight.
Gunfire was also reported overnight in Kostyantynivka and Mariupol as Ukrainian forces tried to reclaim government offices.
Saturday also saw fighting around Kramatorsk, where Ukrainian forces retook the police building, TV tower and SBU security service offices. At least two people were killed in the town.
Our correspondent says passions are running very high following the deaths in Odessa, with pro-Russian activists in Luhansk and Donetsk attacking government buildings.
Pro-Russian gunmen in Mariupol set fire to barricades and videos showed a branch of PrivatBank, a bank owned the pro-Kiev governor of Dnipropetrovsk, being burned down.
Some 42 people died in Odessa on Friday, most of them in the fire at the Trade Unions House, where separatist protesters had barricaded themselves following running battles with pro-Kiev activists.
The blackened building was ringed by police on Saturday and there were some scuffles as groups chanting pro-Russian slogans clashed with government supporters.
The violence there was the most serious in Ukraine since February when more than 80 people were killed during protests in Kiev against the ousted President Viktor Yanukovych.
A Russian government spokesman accused Ukraine's interim government of encouraging nationalist extremists and said it would be "absurd" for the country to hold a planned presidential election on 25 May.
Russia has urged the US to put pressure on Kiev to stop its military operation, which he said risked "plunging the country into a fratricidal conflict".
Washington, in turn, says Moscow should stop backing the pro-Russian separatists - or risk incurring further sanctions from the West.
Ukraine's President Viktor Yanukovych was overthrown by pro-Western protesters in February.
Russia then annexed the Crimean peninsula - part of Ukraine but with a Russian-speaking majority - in a move that provoked international outrage.