Russia has said its forces have entered the north-eastern border town of Vovchansk, near Ukraine's second largest city Kharkiv.
Ukraine's military said it had "pushed the enemy back" from the northern outskirts of the town, adding it had "tactical successes" in several areas.
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 began in February 2022.
Thousands of civilians have fled towards Kharkiv.
There are concerns among Ukrainian commanders about what could happen if Russian troops 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 and has acknowledged Moscow's troops have had some "tactical success".