Authorities in two eastern Ukrainian regions said Saturday that fighting with Russian forces was "fierce", as hope fades for a truce over Orthodox Easter weekend.

The governor of the eastern Kharkiv region, Oleg Sinegubov, said on Telegram that Kyiv retook three villages near the Russian border after "fierce battles". 

"Our units kicked Russian troops out of the settlements of Bezruki, Slatine, Prudyanka," he said, adding that the Ukrainian forced "secured their positions".

He said the battles took place on Friday morning. 

The villages lie north of Ukraine's second city Kharkiv and close to Russian territory. Prudyanka is at around 40 kilometres (25 miles) south-west of the border. 

Sinegubov claimed Russian forces had also attacked residential infrastructure, killing two people. 

Sinegubov said he had agreed with local religious leaders there would be "no night services" in churches on Saturday, the eve before Orthodox Easter, which both Russia and Ukraine celebrate.

"We can't allow crowds, security comes first," he said. 

The governor of the neighbouring Lugansk region, Sergiy Gaiday, said in a video on Telegram on Saturday that fighting also raged there.  

"There is round the clock shelling," Gaiday said, adding that Russian forces "continue to attack" the cities of Rubizhne and Severodonetsk.

He called on people to "evacuate if you have the chance", saying volunteers are helping people leave the area. 

He added, however, that Moscow's forces had "no particular success" and vowed to "chase them, push them out of our land because it is ours".

Presidential advisor Oleksiy Arestovich also claimed Saturday that Russian forces were not making any major breakthrough in their eastern offensive. 

At a briefing he said Russian forces are trying "to develop an offensive on the city of Hulyaipole", in the north of the Donetsk region. 

"They do not manage to do so already for six days," he said. 

Arestovich said Russian forces are "focusing efforts on the area between Kramatorsk and Sloviansk", two cities in the eastern Donetsk region.