Ukrainians and Russians were fighting in nine places along the front line in the Kharkiv area on Saturday morning, the Ukrainian general staff said, adding that they had successfully repelled Russia's offensive.
The latest battles come after the general staff reported that the enemy was deploying "ground troops and technology."
The Ukrainian military has reported Russian advances on two broad sections of the front since Friday, in a widely expected summer offensive, evident in the Kremlin's build-up of many tens of thousands of soldiers near the border.
Russian President Vladimir Putin also threatened an offensive back in March, although military observers in Ukraine, Russia and abroad had not expected the target to be Kharkiv, Ukraine's second-largest city.
The US Institute for the Study of War (ISW) referred to "limited operational objectives" aiming to draw Ukrainian forces away from "other critical sectors" of the front, while bringing Kharkiv back into the range of Russian artillery.
The limited deployment does not suggest "that Russian forces are immediately pursuing a large-scale sweeping offensive operation to envelop, encircle, and seize Kharkiv City, however," the ISW wrote.
Russian troops entered Kharkiv at the very beginning of the war in spring 2022, but were repelled.
According to reports from the front, the attack went in two directions. Russian troops occupied several Ukrainian villages on a section of the border around 30 kilometres north of Kharkiv, in a grey zone ahead of the foremost Ukrainian defences, several reports said. The village of Lipzy was the target of this attack, the Ukrainian general staff said.
The second attack targeted Vovchansk, a town lying on the Russian border some 40 kilometres north-east of Kharkiv. Several smaller towns along the border were also occupied. In Vovchansk, Russia aims to disrupt Ukraine's supply lines towards Kupyansk, observers say.
Meanwhile Ukraine attacked the Russian border region of Belgorod, sending rocket artillery and drones, according to Russian reports. In Belgorod, rocket sirens could intermittently be heard in the morning.
In Rovenki, in the Russian-occupied Luhansk region of Ukraine, shelling caused a fire at an oil depot.
Russia launched its full-scale war against Ukraine more than two years ago. After a rapid advance by the Russians in the first few weeks, Kiev was later able to push back the troops sent from Moscow in some parts of the country.
In the meantime, however, the pendulum has swung back in the other direction - the Ukrainian military is having enormous problems with the supply of weapons, ammunition and soldiers amid daily Russian missile and drone attacks.
Russia occupies around a fifth of Ukraine's territory, including Crimea, which it illegally annexed in 2014.