Rebels in the town, a fierce separatist stronghold where a military helicopter was shot down last week killing 14 servicemen, said they had brought down a Su-25 attack aircraft and a helicopter, but this was denied by Ukrainian authorities.
The fighting in Slaviansk followed a day-long fire-fight on Monday in Luhansk, a town further east on the border with Russia, after an attack by separatists on a border guard camp.
At least two people were killed in the city centre of Luhansk, which like Slaviansk is under separatist control, by a blast which rebels said came from a Ukrainian air strike but which the Ukrainians said was caused by a misfire of a heat-seeking missile by the rebels.
Reports of fresh fighting coincided with Ukraine announcing that 59 servicemen had been killed in clashes with rebels since hostilities broke out in the east of the country in April.
General prosecutor Oleh Makhnitsky said: "In Donetsk and Luhansk regions 181 people have been killed and 293 wounded by terrorist activity, including 59 servicemen."
Ukraine's Russian-speaking eastern regions which border on Russia have been riven with separatist armed rebellions for the past two months.
The Kiev government says the fighting is fomented by Russia, which opposes its pro-Western policies, and accuses Russia of allowing volunteer fighters to cross into Ukraine to fight alongside the rebels.
Vladyslav Seleznyov, a spokesman for the military operation, said: "At the present time the active phase of the anti-terrorist operation is going on near Slaviansk. The separatist fighters are being blocked. If they refuse to lay down their arms they will be destroyed. Our job is to establish peace in the region and this we will do."
He could not confirm reports of wounded on the Ukrainian side around Slaviansk, but he said there were preliminary reports of dead and wounded among the separatist fighters.
Mr Seleznyov added: "Information that Ukrainian planes and helicopters have been shot down are not true. Yesterday one of the helicopters received holes from small arms fire."
Residents in Slovyansk and the nearby cities of Kramatorsk and Krasny Liman were warned to stay at home.
A journalist just south of Slovyansk heard sustained gun and artillery fire and saw plumes of black smoke rising over the city.
Defence analyst Dmytro Tymchuk, who is recognised as having good military sources, said separately that one person had been killed on the Ukrainian side and 13 others wounded when a Ukrainian military column was attacked on its way to Slaviansk from its base in Izyum, to the north-west.
Ukrainian authorities have repeatedly announced an escalation in armed operations, only to eventually back down.
In recent days, government forces have been reinforced to the north of Slovyansk and deployment of air power over the past week has signalled increased determination.