Vladimir Rogov, a Russian-installed official in the region, said offensive actions were concentrated around two towns: Orikhiv, around 50km (30 miles) south of Ukrainian-controlled regional capital Zaporizhzhia, and Hulyaipole, further east.
“The front is mobile, especially in two directions: Orikhiv and Hulyaipole,” Rogov was quoted as saying by the RIA Novosti news agency.
He said there was active fighting in those areas, according to the agency.
“The initiative is in our hands.”
The Russian army later claimed for a second day in a row that it had taken “more advantageous lines and positions” after “offensive operations” in the Zaporizhia region.
It claimed to have hit Ukrainian positions in the village of Lezhyno, outside the regional capital of Zaporizhzhia, which has not fallen to Ukrainian forces.
Al Jazeera was not able to verify the reports.
In its daily report on Sunday, the Ukrainian army said “more than 15 settlements were affected by artillery fire” in Zaporizhia.
Earlier this week, Rogov announced a “local offensive” around Orikhiv and said the Russian army had taken control of the village of Lobkove, near the Dnieper River.
He also said this week that fighting has “sharply increased” in the southern region.
The front in southern Ukraine has been considerably quieter recently than in the east, with Moscow withdrawing from the major city of Kherson in November.
Russia claims to have annexed the Zaporizhia region along with three other Ukrainian regions, but does not control it in its entirety.
While Moscow controls large swathes of the southern part of the region, its main city Zaporizhzhia and northern part are held by Kyiv.
Russia warns the West
The reports come as a Russian official said that governments giving more powerful weapons to Ukraine could cause a “global tragedy that would destroy their countries”.
Vyacheslav Volodin, the speaker of the lower house of Russia’s parliament, warned that countries’ decision to supply Ukraine would lead to a “global catastrophe”.
“If Washington and NATO supply weapons that would be used for striking peaceful cities and making attempts to seize our territory as they threaten to do, it would trigger a retaliation with more powerful weapons,” he said.
Germany is one of the main donors of weapons to Ukraine, and it has ordered a review of its Leopard 2 stocks in preparation for a possible green light.
Nonetheless, the government in Berlin has shown caution at each step of increasing its commitments to Ukraine, a hesitancy seen as rooted in its history and political culture.
Meanwhile, France and Germany committed to showing “unwavering support” to Ukraine during ceremonies and talks Sunday celebrating the 60th anniversary of their post-World War II friendship treaty. In a joint declaration, the countries said they would “stand with Ukraine for as long as it takes”.
They also pledged to “work together for a European Union that is more resilient, more sustainable and more capable to act independently”. The treaty that sealed a bond between longtime enemies France and Germany 60 years ago underpinned today’s EU.
Germany’s tentativeness has drawn heavy criticism, particularly from Poland and the Baltic states, countries on NATO’s eastern flank that were controlled by Moscow in the past and that feel especially threatened by Russia’s renewed imperial ambitions.
Polish Prime Minister Mateusz Morawiecki said that if Germany does not consent to transferring Leopard tanks to Ukraine, his country was prepared to build a “smaller coalition” of countries that would send theirs anyway.