Kostiantynivka, along with other cities, forms a so-called fortress belt in the country's east - an area well fortified by the Ukrainian military.
"We are repelling the Russian occupiers' persistent attempts to gain a foothold in the outskirts of Kostiantynivka using infiltration tactics. Counter-sabotage measures are going on in the city," Oleksandr Syrskyi, Ukraine's army chief, said on Facebook on Saturday.
A Ukrainian battlefield mapping project called DeepState shows that Russian troops control an area around only one kilometre from the city's southern outskirts.
Small chunks of Kostiantynivka, in southeast Ukraine, are marked as a grey zone, meaning neither Ukraine nor Russia has full control over them.
Russia's Defence Ministry said on Wednesday its forces had taken control of Novodmytrivka, just north of Kostiantynivka. Moscow's top general, Valery Gerasimov, said in April that troops were advancing on the north and south of the city.
Syrskyi said that Russian offensive attempts had risen noticeably in April. Since Monday, Russian troops have carried out 83 assaults in this sector using small infantry groups, he added.
Moscow demands that Ukraine pull back from areas in the Donetsk and neighbouring Luhansk regions that Russia has failed to capture during its four-year full-scale war.
US-brokered peace talks stalled over the issue as Ukrainian officials say Kyiv will not cede land it still controls.
Meanwhile, along the northern border with Belarus, Ukraine recorded "rather unusual" activity on Friday, Ukrainian President Volodymyr Zelenskiy said in a post on Telegram on Saturday.
Without elaborating, he said activity was seen on the Belarusian side of the border and that Ukraine would act if matters escalated.
"We are closely documenting and keeping the situation under control. If necessary, we will react," he said.
Belarus, a close ally of the Kremlin, has allowed Russia to use its territory as a staging ground to send troops into Ukraine and to host some of Moscow's tactical nuclear weapons.
For the past few years, Russian troops have not managed to capture any big cities in Ukraine, inching forward and announcing the capture of towns and villages.
The small city of Pokrovsk, whose more than 60,000 pre-war population mostly fled, was the most significant Russian gain in the past year. It took Moscow's troops months to advance, and Kyiv says it still has some positions in the city.
On Saturday, Russia's Defence Ministry said it had seized the village of Myropillia in Ukraine's northern Sumy region, where Moscow says it wants to establish a buffer zone.
But the Kursk group of the Ukrainian military, writing on Facebook, dismissed the Russian report as a "complete lie" and said its units controlled the area.
Also in Sumy, the regional governor said a Russian air strike near the town of Krovelets had injured six people, including two in serious condition.
