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Ukraine's military says it has broken through Russia's first line of defence in the occupied Kherson region.
The reported push appears to form part of a long-awaited counter-offensive being launched by Kyiv in an attempt to retake the country's south.
It follows weeks of Ukrainian attacks aimed at cutting off Russian forces there from main supply routes.
Russia's military claims that Ukrainian troops suffered "heavy losses" during an unsuccessful attacking attempt.
The claims by both Ukraine and Russia have not been independently verified.
Kherson became the first major Ukrainian city to fall into Russian hands after troops advanced into the city from the Crimean Peninsula in the opening days of the war.
Early on Monday, Ukraine's Kakhovka operational group in the south said that one regiment of Russian-backed forces had left its positions in the Kherson region. It added that Russian paratroopers providing the back-up had fled the battlefield.
Oleksiy Arestovych, adviser to President Volodymyr Zelensky's head of office, later also said that Ukraine's armed forces "have broken through the frontline in several places".
In his late night video address, President Zelensky issued a stark warning to Russian forces: "If they want to survive, it is time for Russian soldiers to flee. Go home."
In a briefing early on Tuesday, the presidency said intense fighting was taking place in "almost the entire territory of Kherson", AFP news agency reported.
Explosions and shots were heard for a second day in the regional capital of Kherson. Blasts had earlier been reported in Nova Kakhovka, some 55km (34 miles) north-east and Russia's state-run Ria Novosti news agency reported the city was left without electricity and water supply overnight.
In an update early on Tuesday, a spokesman for Ukraine's operational command in the south said 13 Russian command posts had been destroyed, as well as three ammunition depots and a crossing over the Dnipro river.
Kyiv officials say they have used US-supplied Himars rocket systems to destroy key crossings on the Dnipro in recent weeks and are now targeting makeshift Russian pontoon bridges.
According to Western military sources, Kyiv's strikes are part of a targeted effort to cut off Russian troops on the right (western) bank of the river with the ultimate goal of recapturing the entire Kherson region. A UK defence intelligence assessment said Russia had made significant efforts to reinforce the western bank but most of its units were "likely undermanned and are reliant upon fragile supply lines".
President Zelensky and other top Ukrainian officials have been tight-lipped about details of the reported counter-offensive, urging Ukrainians to be patient.
Responding to the Ukrainian claims, Russia's defence ministry said that Ukrainian troops had attempted an offensive in the Kherson and neighbouring Mykolaiv regions.
The ministry was quoted by Russia's state-run news agencies as saying this operation had failed, and that the Ukrainian troops had "suffered heavy losses".
Analysis by the BBC's Hugo Bachega in Kyiv
Ukraine has long been expected to launch a major offensive to retake Kherson. We could be seeing the beginning of it, although any operation is unlikely to be easy.
Kherson has been under occupation since the early days of the war, and it's one of the largest Ukrainian cities in Russian hands.
For weeks, Ukrainian forces have repeatedly targeted Russian positions deep inside captured territory, away from the front lines.
It's only been possible because of the sophisticated weapons supplied by the West - and it's having a destabilising effect on the invading forces.
The conflict seems to be at a deadlock, with neither side making significant gains. This could be about to change.
Kherson, which had a population of 290,000 before the war, is the only regional capital to have been taken by Russian forces and is currently administered by Moscow-backed officials.
According to Russia's Tass news agency, officials in the Kherson region have started moving forward with plans to hold a referendum on formally joining Russia, prompting accusations by the US that Russia could be preparing to illegally annex parts of occupied southern Ukraine.
Last month, Russia said its military focus was no longer only on eastern Ukraine but on its southern regions of Kherson and Zaporizhzhia too.
Speaking to Russia's RIA news agency, the head of the Russian installed administration, Kirill Stremousov, insisted the city was not under threat of attack and said the "more the area is shelled, the more its inhabitants demand a referendum".
Elsewhere, at least two people were killed and 24 more injured after Russian shelling of the nearby city of Mykolaiv, city officials said.
Russian-installed officials in the Zaporizhzhia region on Tuesday again accused Ukrainian forces of shelling the area around the nuclear power plant. They said on Monday that a hole had been punched in the roof of a fuel depot by a missile strike.
The claim has not been independently verified.
In recent weeks, both Ukraine and Russia have accused each other of shelling Europe's biggest nuclear station, which was seized by Russia in early March. Moscow has kept Ukrainian personnel to operate the station.
Last week, Ukrainian President Volodymyr Zelensky said the world narrowly avoided a radiation accident at the plant, blaming Moscow's actions for this.
An inspection team from the UN nuclear watchdog is expected to arrive at the plant later this week, the organisation's head says.
