Kursk's acting governor Alexei Smirnov says five people were killed on Tuesday, including two members of an ambulance crew, and at least 20 were wounded in the fighting. A senior human rights official said six children were among the injured.
Ukraine made no official comment, though there was evidence of some military action from the Ukrainian side of the border.
Moscow's defence ministry said it sent reserves to help repel hundreds of Ukrainian fighters backed by tanks, in what would be one of the largest ground incursions into Russian territory of the more than two-year-old war.
"The enemy today launched another attempt to break into the territory of Russia's Kursk region," the latest ministry statement said.
"Artillery fire, army aviation strikes and drone strikes are being inflicted on the enemy."
Ukraine regularly fires artillery and missiles into Russian territory and has hit targets deep inside Russia with long-range attack drones, but infantry raids are rare.
Ukrainian fighters backed by tanks launched launched an assault into Russia's southern Kursk region (AP PHOTO)
Forces describing themselves as voluntary paramilitaries fighting on Ukraine's side staged a major incursion into parts of Belgorod and Kursk region earlier this year. The purpose of those raids, which inflicted minimal damage, remains unclear.
Ukraine's general staff on Tuesday reported Russian strikes on border villages but made no mention of any Ukrainian offensive operation inside Russia.
Reuters was unable to verify battlefield accounts from either side.
Russia's defence ministry appeared to have deleted an earlier account of the attack in which it said a "Ukrainian sabotage group" had suffered heavy losses and retreated into Ukraine.
Official Russian social media accounts said that up to 300 Ukrainian fighters, backed by tanks, had attacked border units in two localities in Kursk - Nikolayevo-Daryino and Oleshnya.
Acting governor Smirnov said three Russian civilians had been killed initially, one by shelling and two by drones.
Writing on the Telegram messaging app late in the evening, Smirnov also said Ukrainian forces had shelled a border area, injuring two children.
Officials said the border town of Sudzha had also come under assault and Smirnov said a Ukrainian attack drone had hit an ambulance outside the town, killing the driver and a paramedic and wounding a doctor.
He posted a video telling residents: "I ask you to remain calm and not to be subject to the enemy's information provocations. The situation is controllable."
A senior Orthodox clergyman said Ukrainian shelling had set ablaze a cathedral and other buildings within a large monastery outside Sudzha, but no one was hurt.
Kursk region is also the site of a nuclear power plant, but Smirnov said the facility was operating without incident.
Military authorities in Sumy region - on the other side of the border from Kursk region - said Ukrainian forces had destroyed a Russian ballistic missile, two drones and a helicopter in the region.
Unofficial Ukrainian military blogs showed pictures of what they described as the destroyed helicopter and other equipment.
Ukraine's main military effort is focused on pushing back Russian military forces which control nearly a fifth of its territory. Russian troops have made a series of gradual gains in the past six months.
