"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," Zelensky said in an evening statement.
Mariupol has become a symbol of Ukraine's unexpectedly fierce resistance since Russian troops invaded the former Soviet state 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.’
‘Last chance to save you’
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.
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.
Maksym Khaustov, the head of the Kharkiv region's health department, confirmed the deaths there following a series of strikes that AFP journalists on the scene said had ignited fires throughout the city and torn roofs from buildings.
"The whole home rumbled and trembled," 71-year-old Svitlana Pelelygina told AFP as she surveyed her wrecked apartment. "Everything here began to burn."
"I called the firefighters. They said, 'We are on our way but we were also being shelled.'"
In the eastern city of Kramatorsk, Orthodox Palm Sunday granted its residents some respite before the expected Russian onslaught.
In the Orthodox Svyato-Pokrovsky church, around 40 people -- mostly women wearing colourful headscarves -- attended the service.
"It's very hard and scary right now," said a congregant as she arrived at the red-brick church topped with four gleaming domes.
One young mother, Nadia, said she refused to be evacuated for fear of travelling alone with her two children and leaving her relatives in Kramatorsk.
"We don't go to the basement each time there's a (bomb) siren. It's too stressful for them (the children)," she said.
"We have our spot in the basement just in case, but we prefer to stay in the house if possible. We dim the lights."
And in Kharkiv, the city's metro stations are now home to residents of the eastern metropolis fearful of the battle raging above.
Those impromptu living spaces have become host to makeshift stages, where poets and puppeteers work to lift spirits.
"A person cannot live only with war," Serhiy Zhadan -- a literary celebrity in poetry-obsessed Ukraine -- said.
"It is very important for them to hear a word, to be able to sing along, to be able to express a certain emotion."
Ukrainian deputy prime minister Iryna Vereshchuk has urged Russian forces to let people flee besieged Mariupol, saying that humanitarian corridors allowing civilians to escape would not open on Sunday after failing to agree terms with Moscow forces.
But Lugansk governor Gaiday said he had proceeded with evacuations.
"At our own peril and risk, we took out several dozen people anyway, but it's already dangerous," he told Ukrainian media.
The UN World Food Programme says that more than 100,000 civilians in Mariupol are on the verge of famine and lack water and heating.
And Ukraine's minister of digital transformation Mykhailo Fedorov said the city was on "the verge of a humanitarian catastrophe", saying there was compiling evidence of alleged Russian atrocities there.
"We will hand everything over to The Hague. There will be no impunity."
The mayor of Bucha -- a town near Kyiv where the discovery of dead civilians sparked international condemnation and war crimes accusations -- said Russian troops had raped men as well as women and children there.
Zelensky said he had invited his French counterpart to visit Ukraine to see for himself evidence that Russian forces have committed "genocide" -- a term president Emmanuel Macron has avoided.
"I talked to him yesterday," Zelensky told CNN in an interview recorded on Friday but broadcast Sunday.
"I just told him I want him to understand that this is not war, but nothing other than genocide. I invited him to come when he will have the opportunity. He'll come and see, and I'm sure he will understand."
Zelensky, describing the situation in Mariupol as "inhuman", has called on the West to immediately provide heavy weapons.
But Russia has warned the United States this week of "unpredictable consequences" if it sent its "most sensitive" weapons systems to Ukraine.
Its defence ministry claimed Saturday to have shot down a Ukrainian transport plane in the Odessa region, carrying weapons supplied by Western nations.
On Sunday, spokesman Igor Konashenkov said Russian missiles had destroyed ammunition, fuel and lubricant depots in eastern Ukraine and 44 Ukrainian military facilities, including command posts.
Russian air defence systems shot down two Ukrainian MiG-29 aircraft in the Kharkiv region and a drone near the city of Pavlograd, he added.