The heaviest battles lie ahead in Kherson, as Russian forces dig in to defend the largest city under its control, Kyiv has warned.
Russian troops have been driven back in the region in recent weeks and risk becoming trapped against the west bank of the Dnipro river.
Russian-installed authorities are evacuating residents to the east bank.
But Oleksiy Arestovych, adviser to Ukrainian President Volodymyr Zelenskyy, has said there was no sign that Russian forces were preparing to abandon the city.
“With Kherson everything is clear. The Russians are replenishing, strengthening their grouping there,” Mr Arestovych said in an online video on Tuesday.
“It means that nobody is preparing to withdraw. On the contrary, the heaviest of battles is going to take place for Kherson.”
Kherson is arguably the most strategically important of the four towns Russian president Vladimir Putin proclaimed to have annexed in September. It controls both the only land route to the Crimea peninsula Russia seized in 2014, as well as the mouth of the Dnipro, the vast river that bisects Ukraine.
Yuri Sobolevsky, a member of the ousted pro-Ukrainian Kherson regional council, said the Russia-installed authorities were putting increasing pressure on Kherson residents to leave.
“Search and filtration procedures are intensifying as are searches of cars and homes,” he wrote on the Telegram messaging app.
Meanwhile, Russia took its claim Ukraine is preparing to use a dirty bomb to the UN on Tuesday. The assertion has been dismissed by officials in Kyiv and the West as a false flag for escalating the war.
Russia's Deputy UN ambassador Dmitry Polyanskiy said evidence had been shared with western counterparts.
“I don't mind people saying that Russia is crying wolf if this doesn't happen because this is a terrible, terrible disaster that threatens potentially the whole of the Earth,” he told reporters.
Mr Zelenskyy said Russia's allegation suggested Moscow was planning to use a tactical nuclear weapon and would seek to blame Kyiv.
US President Joe Biden said Russia would be “making an incredibly serious mistake” if it used a tactical nuclear weapon.
Mr Biden later spoke by phone with new British prime minister Rishi Sunak, and they agreed on the importance of supporting Ukraine, the White House said.
In an apparent response to Moscow's allegation, the UN nuclear watchdog said it was preparing to send inspectors to two unidentified Ukrainian sites at Kyiv's request, both already subject to its inspections.
Ukrainian Foreign Minister Dmytro Kuleba told reporters the inspectors would receive full access and called on Moscow to demonstrate the same transparency.
Russia's state news agency RIA has identified what it said were the two sites involved — the Eastern Mineral Enrichment Plant in the central Dnipropetrovsk region and the Institute for Nuclear Research in Kyiv.
Since Russian forces suffered major defeats in September, Mr Putin has doubled down, calling up hundreds of thousands of reservists, announcing the annexation of occupied territory and repeatedly threatening to use nuclear weapons.