A serviceman with a Russian flag on his uniform stands guard near the Zaporizhzhia Nuclear Power Plant. Reuters
Luhansk region Governor Serhiy Gaidai told Ukrainian television, without giving locations, that a "counter-attack is underway and... our forces are enjoying some success. Let's leave it at that".
A presidential adviser had earlier tweeted that there would be "great news" coming from the president on the operation in the northeastern Kharkiv region.
In his evening address, however, President Volodymyr Zelensky made scant mention of operations in the Kharkiv region, but he did say five Russian cruise missiles were shot down on Tuesday, most of them in the south.
Reuters was unable to independently verify the Ukrainian officials' comments and there was no immediate response from Russia.
Ukrainian forces have attacked the Russian-held eastern town of Balakliia in the Kharkiv region.
But an official with the pro-Moscow breakaway Donetsk People's Republic on Tuesday gave an unusually frank account of the fighting at Balakliia, an eastern town of 27,000 people that lies between Kharkiv and Russian-held Izyum, a city with a major railway hub used by Moscow to supply its forces.
"Today, the Ukrainian armed forces, after prolonged artillery preparation ... began an attack on Balakliia... " Daniil Bezsonov said on Telegram, adding that if the town were lost, Russian forces in Izyum would become vulnerable on their northwest flank.
"The Ukrainian armed forces concentrated mass fire on the mobile groups of the Donetsk People's Republic, which had taken up defensive positions in nearby forests.
"At this time, Balakliia is in operative encirclement and within the firing range of Ukrainian artillery. All approaches are cut off by fire."
Several posts on social media from military bloggers and witnesses also reported fighting around Balakliia.
Vadym Krokhmal, member of the town council of Kupyansk, a town east of Kharkiv that has been occupied for the past five months, posted a video online urging residents not to take part in any referendum on joining Russia that occupying forces might stage.
"Very soon, the Ukrainian armed forces will liberate Kupyansk. We know this, we are certain of this," Vadym said, advising people to stock up on food and charge power sources.
"All we need is a little patience."