Large parts of Mariupol have already fallen to pro-Russian forces
The city's Deputy Mayor, Serhiy Orlov, told the BBC that Ukrainian troops there were still fighting.
Tens of thousands of people have died in Mariupol, Ukraine says.
Fighting appears to be continuing around the giant Azovstal steel works in the port, which is one of two areas not under Russian control.
Russian television has broadcast footage which it says shows marines giving themselves up at the steel works.
But an adviser to Ukraine's president insisted that the marines had in fact broken through to connect with Azov battalion forces in another pocket.
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Mariupol is a major port and a key target for Russia as it seeks to establish a land route to the Crimea peninsula, which it annexed in 2014.
The city has been the focus of a devastating assault by Russian forces, and Ukrainian troops there have said they are running out of ammunition.
A senior US defence official said that Russian air strikes continued to target Mariupol and the US does not believe the city has been fully taken by Russia.
Earlier on Wednesday Reuters journalists saw flames billowing from the Azovstal steel works where marines from the 36th brigade have been holed up for weeks.
The population of Mariupol stood at over 400,000 before the Russian invasion. Residents unable to escape the siege have struggled to access the basics for survival.
-- Courtesy of BBC News