Ukraine claims to have retaken more than 1,000 sq km (385 sq miles) of territory in its south and east from Russia in the last week.
BBC reports quoted President Volodymyr Zelensky as saying that the area had been “liberated” as part of a counter-offensive since 1 September.
More than 20 villages in the Kharkiv region have also been recaptured, another general claims.
Russian forces have disputed some of the advances, which if confirmed would mark a significant victory for Ukraine.
Kyiv has liberated “dozens of settlements” near the north-eastern Kharkiv region so far this month, President Zelensky said in his nightly address on Thursday.
In a separate Facebook post Ukraine’s army claimed that in the last three days alone its troops had advanced 50km (31 miles). If verified, that would mark the fastest movement of the front line since the rapid retreat of Russian forces from positions around the capital, Kyiv, in March.
The offensive south-east of Kharkiv would also bring Ukraine’s troops closer to the eastern region of Donetsk, over which Russia has maintained substantial military control since the war began six months ago.
The Kremlin declined on Friday to comment on Kyiv’s claims to have retaken territory near Kharkiv, Reuters news agency reports.
But some Russian officials did dispute the extent of the advances – including Kyiv’s claim to have recaptured the city of Balakliya, made in a video by Ukrainian soldiers and shared by President Zelensky.
Local Moscow-appointed official Vitaly Ganchev told Russian state TV that this was not the case, maintaining that despite a Ukrainian effort to retake the city Russian forces had successfully fought them off.
“The city is under our control,” Mr Ganchev said on Thursday according to Russia’s Tass news agency.
The BBC has not been able to confirm who has control of the city, but verified social media posts show Ukrainian flags flying from administrative buildings in the city’s centre on Thursday.
Ukraine has struggled to push Russian troops back, but these latest advances could position Ukraine’s military within fighting distance of a main railway line supplying Russian forces.
According to analysis by the Institute for the Study of War think tank, Kyiv’s troops are now just 15km (9 miles) from the city of Kupiansk, home to an essential railway junction that Moscow has been using to supply its troops on the battlefield.
While Ukraine claims to have advanced its troops in some parts of the country, in other areas there have been reports of renewed Russian aerial attacks.
On Friday the governor of Ukraine’s north-eastern Sumy region Dmytro Zhvytskyi claimed a Russian air strike had “destroyed” a hospital, wounding multiple people.
The BBC has not been able to verify the report.