Ukrainian troops have set up positions on the east bank of the Dnipro River in southern Kherson region, reports say.
The region is partially Russian-held and crossing the river could be significant in future offensives.
The US-based Institute for the Study of War says Russian military bloggers have posted “enough geolocated footage and text reports to confirm” the advance.
BBC Ukraine says its military sources have reported a “certain movement across [the] Dnipro” near Kherson city.
Ukraine’s military has not confirmed the movement, while Russia has denied the reports.
But if the reports that Ukraine has secured a bridgehead on the east bank are correct, it could be significant in helping Kyiv drive Russian troops back.
A Ukrainian advance in the area could, in the future, even cut the land corridor to Ukraine’s Crimea peninsula, annexed by Russia in 2014.
However, military experts say any Ukrainian troop movements in the bridgehead area – which is crisscrossed by floodplains, irrigation canals and other water obstacles – would be a tough task.
And Ukrainian advances would be further complicated by Russia’s significant advantage in the air.
Ukraine’s military has for some time publicly spoken about preparations for a major counter-offensive, without specifying where and when it could be launched.
Until now, all of the Kherson region on the east bank of the Dnipro has been under Russian control, with the wide river serving as a natural barrier.
The regional capital – sitting on the west bank – was liberated by Ukrainian forces last November.
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In Sunday’s report, the Institute for the Study of War (ISW) said “geolocated footage published on 23 April indicates that Ukrainian forces are operating in areas north-west of Oleshky on the east” bank of Dnipro.
The ISW added there was not enough information to analyse the scale of the reported Ukrainian advance – or the further intentions of the Ukrainian military.
On Monday, Russia’s WarGonzo military blogger reported that Ukrainian troops were “trying to gain a foothold on Bolshoi Potemkin [Velykyi Potyomkin – Ukrainian] island”, which is located between the new and old channels of the Dnipro.
Nataliya Humenyuk, a spokeswoman for Ukraine’s South Command, neither confirmed nor denied reports that Ukrainian forces had secured a bridgehead on the east bank.
She told Ukraine’s TV channels that “difficult work is continuing”.
A military operation requires “informational silence until it is safe enough for our military”, the spokeswoman stressed.
Meanwhile, the Russian-installed head of the Kherson region, Vladimir Saldo, said on Sunday “there were no bridgeheads on the left [east] bank near Oleshky, or any other places” on that side of the river.
In separate developments on Monday:
Moscow’s Vnukovo airport was closed for some time over reports of an identified drone in the area, Russian media said
Russia-installed officials in Sevastopol, Crimea, reported an unsuccessful Ukrainian naval drone attack – Kyiv has not commented
Ukraine’s military said “the fiercest battles” continued in Bakhmut and Mariinka – two key towns in Ukraine’s eastern Donetsk region
The frontlines in southern Ukraine, as they were last month
Source: BBC