Russia's army said it had continued to use aviation, drones to try to put down the attack.
Russia's army appeared to acknowledge Sunday that Ukrainian units had driven deep into the country, saying it had hit troops and equipment around 30 kilometres (20 miles) from the border.
Kyiv launched a surprise border incursion last Tuesday, shocking Russia, which has sent in additional reserves, tanks, drones, artillery and aviation to try to quash the offensive.
The defence ministry in Moscow said Sunday that its troops had "foiled attempts by enemy mobile groups with armoured vehicles to break through deep into Russian territory."
But in an apparent sign of how far some Ukrainian units have managed to advance, it said it had hit enemy units near the villages of Tolpino and Obshchy Kolodez, which are around 25 km and 30 km from the Russia-Ukraine border.
A senior Ukrainian official told AFP that "thousands" of its troops had been deployed to the operation.
Russia's army said Sunday it had continued to use aviation, drones and artillery to try to put down the attack.
It also said it had foiled an attempt by Ukraine to break into another border district of the Kursk region, further to the south from where the bulk of the fighting has occurred so far.
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