The Russian region's governor Vyacheslav Gladkov writes on social media that "the situation at the border is difficult, but under control".
"Our military is performing work according to plan", writes Gladkov, and urges residents to remain calm.
Influential Telegram channels, believed to have good sources within the Russian authorities, had previously written about a Ukrainian attack with several hundred soldiers in the direction of the border town of Shebekino, located in the border areas between the Ukrainian city of Kharkiv and the Russian city of Belgorod.
The information has so far only been reported on the Russian side and has not been commented on from the Ukrainian side.
Ukrainian authorities in war-torn Kharkiv report, however, about Russian artillery fire, where several people have been injured and an elderly man has been killed.
Ukraine's counter-invasion of the Russian region of Kursk, located further north along the border, began three weeks ago. The Ukrainian military claims to have taken control of a total of 100 communities there and a land area of nearly 1,300 square kilometers.
According to the Ukrainian army chief Oleksandr Syrskyj, almost 600 Russian soldiers have been taken prisoner during these weeks.
