Ukraine's military appears to have confirmed reports that its troops have crossed on to the Russian-occupied left (eastern) bank of the Dnipro River.
The armed forces general staff listed Pishchanivka village in the southern Kherson region, 3km (two miles) east of the river, as being shelled by Russia.
The US-based Institute for the Study of War (ISW) said Ukrainian troops had advanced up to 4km east of the river.
Russia's Vladimir Putin insisted that Ukraine's operation was failing.
Ukraine launched its counteroffensive in the south in June, seeking to sever Russia's land corridor to the Crimean peninsula, which Moscow illegally annexed in 2014.
Kyiv aims to reach the Sea of Azov coast, splitting Russian troops in the region in two, and making the Kremlin's supply lines more complicated.
The counteroffensive has so far been slow, bringing only limited territorial gains.
Ukrainian troops have made a number of smaller raids across the Dnipro river before - but the latest reported advance appears to be an attempt to expand the area under their control in anticipation of a larger offensive cross-river operation.
In its report on Thursday morning, the General Staff of Armed Forces of Ukraine said Russia had carried out air strikes on Pishchanivka in the past 24 hours.
It provided no details on whether there were any Ukrainian troops in or near the village.
Hours earlier, the ISW quoted Russian sources as claiming that "likely company-sized elements of two Ukrainian naval infantry brigades conducted an assault across the Dnipro River on to the east bank" on 17-18 October.
"Geolocated footage published on 18 October indicates that Ukrainian forces advanced north of Pishchanivka (14km east of Kherson City and 3km from the Dnipro River) and into [the village of] Poyma (11km east of Kherson City and 4km from the Dnipro River)," the US-based think tank added.
Russian military blogger WarGonzo claimed on Thursday that the Ukrainian units fighting on the eastern bank of Ukraine's main river had been previously trained in the UK.
Meanwhile, the Russian defence ministry appeared to have confirmed Ukrainian operations in the area.
In its report on Wednesday evening, it said Russian troops had "suppressed the activity" of four Ukrainian sabotage and reconnaissance groups in Poyma and the nearby village of Pidstepne.
Moscow claims that, overall, Ukraine's counteroffensive in the Kherson region is failing.
"There is no result yet. There are [Ukrainian] losses," said President Putin during a visit to China on Wednesday.
In other key developments:
Three people were wounded in Russian shelling of Kherson on Thursday, local officials say
Five people have been confirmed dead in the southern city of In Zaporizhzhia in a Russian attack on Wednesday
Further blasts have been reported in Zaporizhzhia and the central city of Dnipro after Russian air strikes. It is not clear whether there are any casualties
