UK intelligence says Kyiv is making "steady gains" around the key city.
Yevgeny Prigozhin said Russia's military had failed in Ukraine.
The Wagner Group claimed victory in the eastern city on Saturday.
Putin's troops have waged a months-long bloody battle – but experts think a rare win would be more symbolic than strategic.
Volodymyr Zelenskyy has warned the “situation is getting more and more difficult” as resistance to Russia's pounding continues.