Kharkiv regional head Oleh Syniehubov said Russian reconnaissance groups had tried to penetrate the border, adding that "not a single metre has been lost".
"Russia has launched a new wave of counteroffensive operations in the Kharkiv sector," said Ukrainian President Volodymyr Zelensky.
Ukrainian commanders have been expecting a summer offensive for some time, possibly even a bid to capture the regional capital Kharkiv. But officials are adamant Russia does not have the resources to do so.
Russia had the capability to aggravate the situation in border areas but not the ability to capture Ukraine's second city, said the head of Ukraine's centre for countering disinformation, Andriy Kovalenko.
Ukrainian reports suggested Russia was trying to create a 10km buffer zone for its Belgorod region, after a series of Ukrainian cross-border attacks.
Friday's small incursions over the Russian border form a familiar yet disturbing axis for Ukrainian forces.
The defence ministry in Kyiv said the attack started with the heavy bombing of the town of Vovchansk "using guided aerial bombs" with the support of artillery. Then, small Russian “scouting groups” moved in across the border, reportedly in several places.
The local head in Vovchansk, 75km (45 miles) north-east of Kharkiv, said the town had come under heavy attack from the early hours of Friday and civilians were being evacuated. Some 3,000 people live in Vovchansk and at least one person was killed and five more injured in the barrage, according to Kharkiv's regional leader.
"At approximately 05:00, there was an attempt by the enemy to break through our defensive line under the cover of armoured vehicles. As of now, these attacks have been repulsed, fighting of varying intensity continues," the defence ministry said.
President Zelensky said the Russians had been engaged "with our troops, brigades and artillery", but added that a fierce battle was under way.
Outwardly at least, officials and generals do not think either of the two regional capitals could fall.
Russia was unable to conquer either city when it had a larger, better-trained force than it does now. Ukrainian sources estimate around 90% of that original 150,000 army are either dead or wounded.
Military commentator Oleksandr Kovalenko has pointed out that Russia needed some 80,000 troops to capture the small eastern city of Avdiivka last February, after months of bombardment. Big cities such as Sumy and Kharkiv were on a completely different scale, he said.
Secondly, Russia has talked about creating a buffer zone between its Belgorod region and Ukraine.
That is because Ukrainian troops have continued to launch artillery strike on Russian territory, to the nervousness of some Western allies.
Civilians were being evacuated from the Vovchansk district while reserve troops move in, officials added.
Moscow has been looking to capitalise on the delayed arrival of American ammunition and weaponry by continuing to push in the eastern Donetsk region.
The return of heavy fighting in the north-east further illustrates Russia's growing confidence and ambitions.
The deputy chief of Ukraine's military intelligence, Maj Gen Vadyn Skibitsky, told The Economist last week that Russia was gearing up for an assault on both Kharkiv and the northern region of Sumy. That warning was repeated by the commander of Ukrainian ground forces, Lt Gen Oleksandr Pavliuk.
Tens of thousands of Russian forces are said to have gathered on the border.
You could be forgiven for seeing a repeat of 2022, when Russia failed to capture Kharkiv and Sumy in the early weeks of its full-scale invasion. Russian forces did occupy the border town of Vovchansk for several months, until they were pushed out in September 2022.