The Russian-installed leader of Ukraine's southern Kherson region, Vladimir Saldo, has called on civilians to evacuate - citing daily rocket attacks by advancing Ukrainian forces, BBC reported.
He urged them to "save themselves" by going to Russia for "leisure and study", and asked for Moscow's help.
His call was later backed up by Russian Deputy Prime Minister Marat Khusnullin in a message on state television.
Ukraine rejects accusations that it targets its own civilians.
Its troops have recently retaken some areas of north-western Kherson, closing in on the regional capital, Kherson city.
"The government took the decision to organise assistance for the departure of residents of the [Kherson] region to other regions of the country," said Mr Khusnullin, who has special responsibility for southern Russia and Crimea.
"We will provide everyone with free accommodation and everything necessary."The first group of people from Kherson would arrive on Friday in Russia's Rostov region, said its governor Vasily Golubev according to Russian state news agency, Tass.
"The Rostov region will accept and accommodate everyone who wants to come to us from the Kherson region," he added.
They have targeted key Russian-held military targets and threatened to cut off the bulk of the occupying forces on the west bank of the Dnieper river (known as Dnipro in Ukraine).
Kherson is the only regional capital seized by Russian forces since Moscow's invasion began on 24 February.
In other major developments on Thursday:
- All of Ukraine - with the exception of Crimea - was for some time under air raid alert, and Russian missile strikes were reported on energy and military targets in the Kyiv region and Lviv, in the west
- Two people were killed in shelling in the southern city of Mykolaiv, and dramatic footage showed a young boy being rescued from the rubble of a destroyed house, although he later died, officials said
- Both Kyiv and Moscow confirmed that 20 Ukrainian service personnel were exchanged for 20 Russian soldiers - in the latest such swap
- Russia accused Ukraine of hitting a residential building in the Russian border city of Belgorod
- Russian President Vladimir Putin met his Turkish counterpart Recep Tayyip Erdogan, and proposed building a gas hub in Turkey as an alternative supply route to Europe following problems with the Nord Stream pipelines
- Nato said it would provide Ukraine with dozens of jammers - transmitters used to disrupt signals - to counteract Russian and Iranian drones. The head of the military alliance, Jens Stoltenberg, also said members had agreed to increase protection of critical infrastructure after what he called the "sabotage" of the Nord Stream pipelines
Speaking on Thursday, Mr Saldo said many towns in the region - including the two major cities of Kherson and Nova Kakhovka - were now under daily rocket attacks by Ukrainian troops.
"Such strikes are causing serious damage," he said, urging residents across the whole region - and especially those on the west bank of the Dnieper river - to evacuate to Russia or Crimea, BBC reported.
And he appealed to the government in Moscow to help organise the process. "Russia is not abandoning its people," he stressed, using a popular saying.
