Ukraine said Monday that its troops were advancing around Bakhmut while playing down the importance of its "offensive actions", after Russia said it had repelled a large-scale attack.
"The Bakhmut sector remains the epicenter of the hostilities. We are advancing there on a rather wide front," Ukrainian Deputy Defense Minister Ganna Malyar said.
"We are having some success."
The now largely destroyed city of Bakhmut in the Donetsk region in eastern Ukraine has been the scene of the longest and one of the bloodiest battles of the war.
The contrasting claims came as Pope Francis's peace envoy, Italian Cardinal Matteo Zuppi, arrived in Kyiv for two days of talks.
Ukraine says it has been preparing a major offensive, after months of stalemate, to recapture territory lost since Russia began its invasion of the pro-Western country in February last year.
But Kyiv, which has been reinforced by supplies of advanced weapons from its allies, said there would be no formal announcement about the start of the offensive.
On Sunday, Ukrainian Defense Minister Oleksiy Reznikov posted a cryptic tweet, quoting lyrics from Depeche Mode's song "Enjoy the Silence".
"Words are very unnecessary," he tweeted. "They can only do harm."
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 defenses for weaknesses before starting a full-blown offensive.
Early on Monday, Russia's defense ministry said "the enemy launched a large-scale offensive in five sectors of the front" on Sunday in the south of the Donetsk region.
"A total of six mechanized and two tank battalions of the enemy were involved," it said in a Telegram post.
"The enemy did not achieve their tasks. They had no success."
The ministry posted what it said was a video of the battle, showing Ukrainian armored vehicles coming under heavy fire.
Putin's top commander in Ukraine, Chief of the General Staff Valery Gerasimov, "was at one of the advanced command posts", the ministry said.
Both 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, which mercenary group Wagner last month claimed had fallen to Moscow.
Kots suggested Kyiv had not yet "introduced the main forces into battle".
Pro-Moscow war blogger Alexander Khodakovsky said Ukrainian troops had deployed Leopard tanks in the east.
Wagner boss Yevgeny Prigozhin said on Monday that Ukraine's troops had made gains near Bakhmut.
Russian troops, he said, were "slowly" leaving the village of Berkhivka near Bakhmut.
"Disgrace!", he said, challenging Russian Defense Minister Sergei Shoigu to come to the front.
"Come on! You can! And if you can't, die a hero," he said.
A Moscow-installed official said Ukrainian troops were on the offensive in the southern region of Zaporizhzhia, home to Europe's largest nuclear plant, which has been partly under the control of Russia's forces 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.
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.
Speaking to reporters in New Delhi, US Secretary of Defense Lloyd Austin said the Ukrainian authorities would make an announcement "at the right time".
"We can talk about what we're seeing -- and we're seeing continued operations in and around the Bakhmut area," he added.
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.
Ukraine has consistently denied responsibility for cross-border attacks on Russian soil.
Fighting on Russia's border with Ukraine has forced thousands of residents to flee.
In the latest development, Russian authorities were forced to issue denials on Monday after several radio stations were hacked and played a fake President Vladimir Putin speech announcing an invasion by Kyiv's troops and emergency measures in three border regions.