Russian forces are shelling the port of Mykolaiv in southern Ukraine, according to the regional governor.
Vladimir Putin’s troops are on the northeast side of the Black Sea city but have not taken it despite increasing pressure, said a senior US defence official.
If they capture Mykolaiv, then it will open up the route to Odesa, seen as one of Russia’s next strategic goals, about 80 miles away, or a two-hour drive.
It comes as Odesa’s mayor said Russian troops could soon surround his city on three fronts.
Mayor Gennadiy Trukhanov claimed Mr Putin’s forces would seek to advance from areas they have occupied in the Mykolaiv region towards Moldova’s separatist-controlled Transnistria region, where Russian troops are based.
Such a move could cut Odesa off from the rest of Ukraine.
“Also, we think that while this is happening, Russian landing ships could surround us from the sea,” Mr Trukhanov added.
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Ukraine war: Gun battle on outskirts of Kyiv, video shows – as Russian forces make renewed advance towards capital
In the south of the country, Russian shelling again prevented evacuees leaving the port city of Mariupol, according to Ukrainian deputy prime minister Iryna Vereshchuk.
Mr Putin’s forces have laid siege to Mariupol, with Ukraine saying 1,582 civilians have died there in 12 days following the invasion.
Ms Vereshchuk also said Russian forces stopped some buses of people trying to flee the Kyiv region.
But she added some evacuations were successful, including 1,000 people who left the village of Vorzel, also in the Kyiv region.
In Melitopol, which is under Russian control, Mr Putin’s forces have allegedly violated international law by abducting the southern city’s mayor.
Ukrainian officials at the foreign ministry say the troops kidnapped Ivan Fedorov after wrongly accusing him of terrorism.
Moscow has not commented on what happened to Mr Fedorov.
Russian forces appeared to be making a renewed offensive toward Kyiv, with Ukrainian troops engaging in a firefight on the outskirts of the capital.
Video footage appeared to show Ukrainian troops, carrying RPGs and anti-tank weapons, taking cover before firing into a wooded area in a village near Kyiv.
Ukrainian President Volodymyr Zelenskyy has said his country will win the war against Russia – while Kyiv’s leaders said people who had fled were coming back to fight as Mr Putin’s troops edged closer to the capital.
The 44-year-old, who has refused to leave Kyiv, vowed not to give “a single piece of land” to Russia, in a defiant speech to his people.
After days of being stalled, Russia’s military could be as close as nine miles (15km) from the city centre, according to a senior US defence official.
As well as advancing on the northwest route into Kyiv, troops are approaching along two parallel lines east of the capital.
Chernihiv, northeast of the capital, is now isolated but the Russian military has not moved past this city, with heavy fighting reported as Ukrainians fight back.