Ukrainian President Volodymyr Zelensky on Monday praised his troops for advances claimed near the devastated city of Bakhmut, while Russia said it had repelled a large-scale attack.
The contrasting claims came as Pope Francis's peace envoy, Italian Cardinal Matteo Zuppi, arrived in Kyiv for two days of talks.
Earlier, Ukrainian Deputy Defence Minister Ganna Malyar had noted "some success" on the battlefield.
"The Bakhmut sector remains the epicentre of the hostilities. We are advancing there on a rather wide front," she said.
"The defensive operation includes counteroffensive actions. Therefore, in some sectors, we are conducting offensive actions," Malyar added.
In May, Russia said it seized the now-destroyed eastern Ukrainian city of Bakhmut, the scene of the longest and one of the bloodiest battles of the war.
Ukraine says it has been preparing a major offensive to recapture territory lost to Russia, but that there would be no announcement about when it would start.
US President Joe Biden wished Ukraine well on Monday in its expected counteroffensive.
Asked by AFP if he thought the expected Ukrainian pushback would work, the president answered by silently raising his hand and crossing his middle and index fingers.
The war has escalated in recent weeks, with increased attacks on both sides of the border with Russia.
Military experts expect Ukrainian forces to test Russian defences for weaknesses before starting a full-blown offensive.
- 'Large-scale offensive' -
Earlier on Monday, Russia's defence ministry said "the enemy launched a large-scale offensive in five sectors of the front" on Sunday in the south of the Donetsk region.
Moscow said it had thwarted the offensive, killing a total of "1,500 servicemen" and destroying more than 100 armoured vehicles.
But the claims were dismissed on Tuesday by the boss of Russian mercenary group Wagner, Yevgeny Prigozhin, as "wild fantasies".
Prigozhin has been embroiled in a public spat with Russia's regular army and has accused Moscow's military leadership of not supplying enough ammunition, among other criticisms.
Large parts of Donetsk have been held by pro-Moscow separatists since 2014.
It is one of four eastern Ukrainian territories that Russia formally annexed in September last year, along with Lugansk, Zaporizhzhia and Kherson, despite never fully controlling them.
The ministry posted what it said was a video of the battle, showing Ukrainian armoured vehicles coming under heavy fire.
Russian President Vladimir Putin's top commander in Ukraine, Chief of the General Staff Valery Gerasimov, "was at one of the advanced command posts", the ministry said.
The ministry later said it defeated "new attempts to pierce Russian defences" in southern Donetsk.
The Russian and Ukrainian claims could not be independently verified.
A high-profile Russian war correspondent, Alexander Kots, said "battles have been going on" around Vugledar, in the south of the Donetsk region, and further north in Soledar and Bakhmut, which were occupied by Moscow's forces after months of fighting.
Kots said Ukrainian forces were "conducting offensive operations" in and around Bakhmut but suggested Kyiv had not yet "introduced the main forces into battle".
- Offensive 'does not exist' -
Prigozhin also said Monday that Ukraine's troops had made gains near Bakhmut.
Russian troops, he said, were "slowly" leaving the village of Berkhivka near Bakhmut, which he said was a "disgrace".
In the southern region of Zaporizhzhia, a Moscow-installed official said Ukrainian troops were on the offensive.
The region is home to Europe's largest nuclear plant and has been partly under Russian control since the start of Moscow's offensive.
"It looks like the die has been cast and the next couple of months will clear up a lot. The fight will be serious because there is a lot at stake," Vladimir Rogov said.
Ukrainian presidential aide Mykhaylo Podolyak said: "Moscow is actively involved in repelling... a global offensive that 'does not yet exist'."
- Pope's envoy in Kyiv –
The Vatican said Pope Francis's peace envoy Zuppi, the head of the Italian bishops' conference, had headed to Kyiv on Monday for talks on the war with the Ukrainian authorities.
Ukraine's ambassador to the Vatican, Andrii Yurash, said on Twitter he hoped Zuppi's visit would help find "appropriate answers" for a peaceful solution.
But attacks have, in fact, intensified in recent weeks, with the latest and most striking being the regular shelling and cross-border incursions from Ukraine in Russia's southern Belgorod region.
The border breaches have been claimed by Russian anti-Kremlin ultra-nationalists.