An online Ukrainian news website, Insider, said eight pro-Russian militants had been killed.
Russia's Interfax news agency said Ukrainian troops used large caliber weapons as they tried to take an occupied government building in the city.
Members of a crew from Britain's ITV News in the city said on Twitter that they saw heavy guns being used and smoke from burning tires rising above Mariupol.
"Just filmed serious fighting downtown Mariupol. Snipers everywhere. Heavy guns. A man got shot just next to me. Black smoke over the city," wrote cameraman Daniel Demoustier.
Ukrainian military armored vehicles were seen in the center of Mariupol.
The fighting occurred as Ukraine commemorated Victory Day celebrations marking the Soviet defeat of German forces in World War II.
Kiev held low-key ceremonies while Russia put on a display of military might in Moscow's Red Square.
NJF/NJF