Russia will pull out its forces from the Ukrainian towns of Balakliia and Izium, the state-run news agency TASS quoted the Russian defence ministry as saying on Saturday.
Kyiv had announced recently that its troops had made sweeping gains in the territory held for months by Russia's army.
"In order to achieve the stated goals of the special military operation to liberate the Donbas (region in Eastern Ukraine), it has been decided to regroup the Russian troops stationed in the districts of Balakliia and Izium," said Major General Igor Konashenkov, spokesman for the defence ministry, according to TASS.
A witness in Valuyki, a town in Russia's Belgorod region near the border with Ukraine, told Reuters she had seen scores of people who had fled from the Ukrainian town of Kupiansk, another key supply base.
She said families were eating and sleeping in their cars along roads.
"I was at the market today and saw a lot of people from Kupiansk. They say half of the city was taken by the Ukrainian army and Russia is retreating ... the fighting is getting closer," the witness said.
Denis Pushilin, leader of the self-proclaimed Donetsk People's Republic in eastern Ukraine, much of which is occupied by Russian forces, acknowledged "problems" in another part of the frontline.
He said the situation in the town of Krasny Liman "remains quite difficult - as in a series of settlements in the north of the republic".
