The statements from national security council secretary Andriy Parubiy came as no new peace talks were in sight four days after a cease-fire expired.
Russia and European leaders have been calling for talks with the rebels to begin by Saturday, but no meeting has yet been announced.
Parubiy said government forces were attacking rebel positions with artillery and planes and that 17 villages had been recaptured since a unilateral cease fire expired on Monday.
"Fighting aiming at defending the border and liquidating terrorists in the Luhansk and Donetsk regions continues," Parubiy told journalists at a news conference. "This is taking place during the constant movement of military equipment and armed forces of the Russian Federation close to the Ukraine's border."
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He said Ukrainian border checkpoints in the Luhansk and Donetsk regions are still being attacked and those attacks are coming both from inside Ukraine and from Russian territory.
Rebels shelled the Donetsk airport overnight and an AP reporter said a radar or communications tower there was still burning in the morning.
Ukraine says Russia is arming and supporting the rebels, a charge that Russia denies. Ukraine is also wary that Russia could try to grab more territory after it annexed the peninsula of Crimea in March.
The rebels have asked to join Russia and Russian nationalists have urged President Vladimir Putin to send troops into Ukraine, but Putin has resisted those demands for fear of more sanctions.