Russia’s defense minister, Sergei Shoigu, told Mr Putin on Thursday that the sprawling Azovstal steel factory where Ukrainian forces are holed up is “securely blocked”.
He said the rest of the city has been “liberated”, which Mr Putin hailed as a “success”.
Leaving the factory in Ukrainian hands, however, robs the Russians of the ability to declare a complete victory in Mariupol.
The city’s capture has both strategic and symbolic importance.
Meanwhile, the Kremlin said it submitted a draft of its demands for ending the war as the West raced to supply Ukraine with heavier weapons to counter the Russians’ new drive to seize the industrial east.
Britain’s Ministry of Defense (MoD) said in an assessment on Thursday that Russia likely wants to demonstrate significant successes ahead of its annual May 9 Victory Day celebrations.
“This could affect how quickly and forcefully they attempt to conduct operations in the run-up to this date,” it said.
With global tensions running high, Russia reported the first successful test launch of a new type of intercontinental ballistic missile, the Sarmat.
President Vladimir Putin boasted that it can overcome any missile defense system and make those who threaten Russia “think twice”.
The head of the Russian state aerospace agency called the launch out of northern Russia “a present to Nato”.
The Pentagon described the test as “routine” and said it was not considered a threat.
On the battlefield, Ukraine said Moscow continued to mount assaults across the east, probing for weak points in Ukrainian defensive lines.
Russia said it launched hundreds of missile and air attacks on targets that included concentrations of troops and vehicles.
The Kremlin’s stated goal is the capture of the Donbas, the mostly Russian-speaking eastern region that is home to coal mines and metal and heavy-equipment factories.
Detaching it from the rest of Ukraine would give Mr Putin a badly needed victory two months into the war, after the botched attempt to storm the capital, Kyiv.
The MoD said Russian forces were advancing from staging areas in the Donbas toward Kramatorsk, which continues to suffer from persistent rocket attacks.
In a video address, Ukraine President Volodymyr Zelensky said the Russians were not “abandoning their attempts to score at least some victory by launching a new, large-scale offensive”.
The Luhansk governor said Russian forces control 80% of his region, which is one of two that make up the Donbas.
Before Russia invaded on February 24, the Kyiv government controlled 60% of the Luhansk region.
Governor Serhiy Haidai said the Russians, after seizing the small city of Kreminna, are now threatening the cities of Rubizhne and Popasna.
I have urged all residents to evacuate immediately.
“The occupiers control only parts of these cities, unable to break through to the centres,” Mr Haidai said on the messaging app Telegram.
Analysts have said the offensive in the east could become a war of attrition as Russia faces Ukraine’s most experienced, battle-hardened troops, who have fought pro-Moscow separatists in the Donbas for eight years.
Ukraine-Russia: Holocaust survivor, 91, dies in basement in besieged Mariupol
Russia said it presented Ukraine with a draft document outlining its demands for ending the conflict — days after Mr Putin said the talks were at a “dead end”.
Kremlin spokesman Dmitry Peskov said “the ball is in their court, we’re waiting for a response”.
He gave no details on the draft and it was not clear when it was sent or whether it offered anything new to the Ukrainians, who presented their own demands last month.
Mr Zelensky said he had not seen or heard of the proposal, though one of his top advisers said the Ukrainian side was reviewing it.
Moscow has long demanded Ukraine drop any bid to join Nato.
Ukraine has said it would agree to that in return for security guarantees from other countries.
Other sources of tension include the status of both the Crimean Peninsula, seized by Moscow in 2014, and eastern Ukraine, where the separatists have declared independent republics recognized by Russia.
In devastated Mariupol, Ukraine said the Russians dropped heavy bombs to flatten what was left of the sprawling Azovstal steel factory, believed to be the city’s last pocket of resistance.
A few thousand Ukrainian troops, by the Russians’ estimate, remained in the factory and its labyrinth of tunnels and bunkers spread out across about four square miles.
Mr Zelensky said about 1,000 civilians were also trapped.
A Ukrainian apparently in the facility posted a Facebook video urging world leaders to help evacuate people, saying: “We have more than 500 wounded soldiers and hundreds of civilians with us, including women and children.”
The officer identified himself as Serhiy Volynsky of the 36th Marine Brigade and warned: “This may be our last appeal. We may have only a few days or hours left.”
The authenticity of the video could not be independently verified.
The Russian side issued a new ultimatum to the defenders to surrender, but the Ukrainians have ignored all previous demands.
More than 100,000 people overall were believed trapped in Mariupol with little if any food, water, medicine or heat.
The city’s pre-war population was 400,000.
Ukrainian deputy prime minister Iryna Vereshchuk said the latest effort to open a safe corridor for women, children and the elderly to escape failed because the Russians did not observe a cease-fire.
Many previous such agreements have failed apart because of continued fighting.
An adviser to Mr Zelensky, Mykhailo Podolyak, said on Twitter that he and other Ukrainian negotiators were ready to hold talks without conditions to save the lives of trapped Mariupol defenders and civilians.
There was no immediate response from Russia.
Mariupol holds strategic and symbolic value for both sides.
The scale of suffering there has made it a worldwide focal point of the war.
Mariupol’s fall would deprive Ukraine of a vital port, complete a land bridge between Russia and the Crimean Peninsula, and free up Russian troops to move elsewhere in the Donbas.
Elsewhere, some residents of the eastern city of Kharkiv have lived in bases for weeks, trying to stay safe from Russian shelling.
With no running water, gas or electricity, they collect rainwater and cook on open fires, burning debris from destroyed wooden buildings.
In one neighborhood, they sought safety in a school basement — using desks, tables and chairs to fashion beds.
More than 300 people slept there during the first days of the war, but most have left for safer places, and only a few dozen remain.
Stirring a large pot of a thin vegetable soup, a woman said volunteers brought cabbage, beetroot and canned beans.
“We mixed everything together and made borscht,” said Natasha, who only gave her first name.
As Russia funneled troops and equipment into the Donbas, western nations rushed to increase the flow of military supplies to Kyiv for this new phase of the war — likely to involve trench warfare, long-range artillery attacks and tank battles across relatively open terrain.
US President Joe Biden was set to announce plans on Thursday to send more military aid to Ukraine, according to a US official who was not authorized to comment publicly and spoke on the condition of anonymity.
Mr Zelensky said Ukraine’s western allies have “come to understand our needs better”, adding that Ukraine is receiving new shipments of western weapons “now, when Russia is trying to step up its attacks, not in weeks or in a month”.
Mr Putin, meanwhile, boasted that the Sarmat missile has “no equivalents in the world”.
The Sarmat is intended to eventually replace the Soviet-built missile code-named Satan by Nato as a major component of Russia’s nuclear arsenal.
It will ”make those who, in the heat of frantic, aggressive rhetoric, try to threaten our country think twice”, Mr Putin said.