Russia is claiming its forces have entered the north-eastern border town of Vovchansk, near Ukraine’s second largest city Kharkiv.
Ukraine says there is still heavy fighting taking place there.
Russia has intensified its attacks on the region following Friday’s surprise incursions across the border, seizing at least nine villages and settlements in one of the most significant ground attacks since its full-scale invasion of Ukraine in February 2022.
Thousands of civilians have fled towards Kharkiv, and there are concerns among Ukrainian commanders about what could happen if Russian forces get within artillery range of the city.
Ukraine’s army has said that Russia has deployed “significant forces” – up to five battalions – in its latest offensive, acknowledging it had had “tactical success”.
But in a statement, the army claimed that Russia had lost more than 100 troops in a day, adding that it was taking steps to repel advancing forces.
Vovchansk, located 74km (45 miles) from Kharkiv, has been heavily bombed in recent days and officials in the surrounding region say Moscow is now targeting settlements with glide bombs.
While it is a significant town in the region, Vovchansk is not of specific military importance, though its capture would be a blow to Ukrainian morale.
Kharkiv regional head Oleh Syniehubov said Russia was deliberately trying to stretch the front line by attacking in small groups in new directions.
He told local television that Ukrainian forces were holding Russian troops back but warned that fighting could spread to new settlements, according to Reuters.
Nearly 6,000 people have been evacuated in the region, he said, adding that 30 settlements had been struck by mortar or artillery shelling.
While it is thought this Russian cross-border incursion is still unlikely to be able to take Kharkiv, their gradual approach on this new, north-eastern front line only increases the anxiety for those living there.
Frustrated Ukrainian troops have told the BBC that Russian soldiers were initially able to just “walk in” to the region because of the lack of defences.
Kostyantyn Tymchenko, who lives in Vovchansk, is among those leaving the town for the relative safety of Kharkiv and said he was shocked by how close the fighting was.
“Half a kilometre away, there is already fighting, automatic weapons,” he said.
“On the one side [of the Vovcha River] are [Russians], on the other – ours.
“Tanks are constantly approaching, shooting back and then leaving. I thought it would be okay. I was shocked. I wish I had known in advance.”
Away from Kharkiv, two people were killed by shelling in the Russian-controlled city of Donetsk in eastern Ukraine, the Russian-installed mayor has claimed.
And at least three people were killed by a strike on an industrial zone in Krasnodon, known as Sorokyne in Ukrainian, a Russian-held part of eastern Ukraine, the Moscow-installed local governor said.
Ukraine has not commented on these incidents.
Russia also said it had shot down 31 Ukrainian drones over several regions in annexed-Crimea.
It comes after Russian officials said at least 15 were killed when a section of an apartment block fell in the Russian border city of Belgorod when it was struck by fragments of a Ukrainian missile which had been shot down.
A Ukrainian security source said Ukrainian forces had struck an oil terminal and electrical substation in western Russia.