Ukrainian President Volodymyr Zelensky has accused Russia of wanting to "destroy" the entire eastern region of Donbas.
Moscow is pushing for a major victory in Mariupol as it seeks to gain control of Donbas and create a land corridor to Crimea, which it annexed in 2014.
Ukrainian troops, however, have pledged to continue fighting in defence of the city. It comes after Russian forces called on the remaining fighters in the encircled Azovstal steel plant to surrender.
Ukrainian authorities have urged civilians in Donbas to move west and escape a large-scale Russian offensive to capture the Donetsk and Lugansk regions (within Donbas).
President Zelensky said in an evening statement: "Russian troops are preparing for an offensive operation in the east of our country in the near future. They want to literally finish off and destroy Donbas."
Mariupol has been a symbol of Ukraine's fierce fightback since Russian troops invaded the country on 24 February.
"The city still has not fallen," Prime Minister Denys Shmyhal said.
"There's still our military forces, our soldiers. So they will fight to the end," he told ABC's This Week.
"We will not surrender."
While several large cities were under siege, he said, not one - with the exception of Kherson in the south - had fallen, and more than 900 towns and cities had been re-captured.
Lugansk governor Sergiy Gaiday said the coming week would be "difficult".
"It may be the last time we have a chance to save you," he wrote on Facebook.
Russian forces continued to shell the eastern Lugansk region and two people died in the town of Zolote, he told Ukrainian media earlier in the day.
Five "powerful" Russian missile strikes hit Lviv early Monday, in a rare attack on the western city, which has so far been spared much of the fighting.
Two people also died and four were wounded in attacks on the towns of Marinka and Novopol, west of Donetsk - regional governor Pavlo Kyrylenko said on Telegram - and an air strike hit an armaments factory in the capital Kyiv.
In the country's second city of Kharkiv, at least five people were killed and 20 wounded in a series of strikes just 21 kilometres (13 miles) from the Russian border.