Ukraine's top military command said on Monday its forces were engaged in heavy battles in frontline hot spots, a day after Kyiv said it had made the first modest gains in reclaiming territory from Russia as part of its counteroffensive.

Some 25 battles had taken place over the past day near the eastern town of Bakhmut, and further south near Avdiivka and Maryinka, all in the Donetsk region, but also near Bilohorivka in the Luhansk region, Ukraine's armed forces general staff said.

Ukraine said on Sunday its troops had made advances on three villages in Donetsk: Blahodatne, Neskuchne and Makarivka.

The claims could not be independently verified and there was no immediate comment from Russian officials.

Some prominent Russian military bloggers indicated that while Ukrainian forces took Blahodatne and Neskuchne, fighting for Makarivka was going on.

Both sides have said their forces had inflicted heavy personnel and equipment losses on their opponents over the past week when Ukraine's counteroffensive was taking shape.

Russian President Vladimir Putin said on Friday that Ukraine had failed to breach Russian defences, while his defence ministry said it had destroyed several main battle Leopard tanks and other equipment that Ukraine had received from the West.

While staying largely silent over the past week about its counteroffensive, Ukraine's military has reported daily battlefield successes.

"Over the last week in the Bakhmut direction, the Russian invaders suffered significant losses," the general staff said on Monday.

The governor of the Donetsk region in Ukraine said one civilian was killed and two were wounded by Russian fire in the Avdiivka region on Sunday.

Ukrainian Defence Minister Oleksii Reznikov said that Kyiv wanted to discuss details of the "aircraft coalition" with its allies at the next meeting of the Ukraine Defence Contact Group on June 15 in Brussels.

Zelenskiy has long been appealing for the U.S.-made F-16 fighter jets, saying their acquisition with Ukrainian pilots would be a sure signal from the world that Russia's invasion would end in defeat.

"At this stage, we are talking about training of pilots ... and our technicians and engineers," Ukraine's Military Media Centre quoted Reznikov as saying.