Heavy fighting in Russia's Kursk region has entered a third day, with "ongoing" efforts to expel Ukrainian forces from the country, Moscow's defence ministry has said.
Russia says at least 1,000 Ukrainian troops, supported by tanks and armoured vehicles, crossed the border on Tuesday.
Some 3,000 people have had to evacuate the region, as ongoing military altercations have killed at least four people, Kursk's deputy Governor Andrei Belostotsky said on Thursday.
Ukraine's military is yet to directly comment on the reports, but President Volodymyr Zelensky said, without referencing the assault: “Everyone can see that the Ukrainian army knows how to surprise [and] achieve results.”
Mykhailo Podolyak, a long-term aide to President Zelensky, wrote on X, formerly Twitter: "War is war, with its own rules, where the aggressor inevitably reaps corresponding outcomes."
In a televised meeting of government officials on Wednesday, Russian President Vladimir Putin accused Ukraine of launching a "major provocation" and "firing indiscriminately" at civilian buildings and residences.
Russia's Defence Ministry said "breakthrough attempts" by the Ukrainian army's formations in the Sudzha and Korenevo districts in Kursk region had been thwarted in a combined effort by the Federal Security Service (FSB) and the military.
Mr Belostotsky claimed Kyiv's forces were beginning to retreat from the region.
On Wednesday, Russia's Chief of General Staff Valery Gerasimov said the "advance" into the Kursk region had been stopped, with Russian forces "continuing to destroy the adversary in areas directly adjacent to the Russian-Ukrainian border".
While the FSB made claims to a similar effect on Tuesday, when initial reports of a potential Ukrainian incursion surfaced.
In its latest report, the Institute for the Study of War, a well-respected think tank, said geo-located footage from the past two days showed that Ukrainian armoured vehicles had advanced to positions 10km (6.2 miles) into the Kursk region.
Meanwhile, footage verified by the BBC has shown fighter jets flying low overhead in the region in recent days.
Ukraine’s key allies seemed surprised by the offensive - with White House press secretary Karine Jean-Pierre saying the US was reaching out to Ukraine for more information.
Russia's military leadership is under severe scrutiny domestically, as some popular and generally well-informed pro-war Telegram channels have said the situation on the ground is not as stable as the Kremlin has suggested.
The influential pro-war Rybar Telegram channel went on to strongly criticise the highest ranks of the Russian military on Wednesday, saying that "for two months the full information was sent to the useless headquarters", adding that there was enough time "to make an appropriate decision".
Rybar's concerns were echoed by several other pro-war bloggers.
Local leaders in regions adjacent to Kursk, both in Russia and in Ukraine, ordered residents to evacuate the area since news of the incursion.
On Wednesday, the head of the Ukrainian region of Sumy, Volodymyr Artyukh, ordered the evacuation of areas bordering Kursk.
While in the Russian region of Belgorod, governor Vyacheslav Gladkov said on Thursday that settlements across his province were attacked by Ukrainian forces over the past 24 hours.
This is not the first incursion into Russia by fighters based in Ukraine. Some groups of anti-Kremlin Russians launched raids last year, which were repelled.
The forces crossed into the Belgorod and Kursk regions again in March, where they engaged in clashes with Russian security forces.
On Wednesday evening, Ukrainian MP Oleksiy Honcharenko said the Ukrainian army had established control over the Sudzha gas hub, a major gas facility involved in the transit of natural gas from Russia to the EU via Ukraine, which has continued despite the war.
It is the only point of entry for Russian gas into the EU.
But on Thursday gas was reportedly still flowing from Sudzha.
Russia's National Guard said it had reinforced security around Kursk's nuclear power station, which lies some 70km northeast of Sudzha.
For the past few months Russia has made incremental gains in eastern Ukraine, as many of Kyiv's ground forces have faced relentless attacks in the eastern Donbas region.
In a recent interview Ukraine's head of defence intelligence, Kyrylo Budanov, said the main offensive by Russian forces "is expected to be over in a month and a half to two months".