Ukrainian troops have advanced up to 10 kilometres (six miles) into Russia in possibly the most serious border incursion of the conflict, the Institute for the Study of War said on Thursday.
Pro-Kyiv forces stormed into Russia's southwestern Kursk region on Tuesday morning, deploying around 1,000 troops and more than two dozen armoured vehicles and tanks, according to the Russian army.
Kyiv has not officially commented on the operation.
Russia's top general vowed on Wednesday to crush the incursion and push the Ukrainian fighters back to the border.
"Ukrainian forces have made confirmed advances up to 10 kilometres (six miles) into Russia's Kursk Oblast amid continued mechanised offensive operations on Russian territory," the US-based Institute for the Study of War (ISW) said in its latest update.
"The current confirmed extent and location of Ukrainian advances in Kursk Oblast indicate that Ukrainian forces have penetrated at least two Russian defensive lines and a stronghold," it added.
The advance has centred on the logistical hub of Sudzha, a town of around 5,000 inhabitants located eight kilometres from the Ukrainian border.
There have been scant updates from officials in Moscow.
But Russian military bloggers, who have links to the army, also reported Kyiv had made significant advances and lamented deteriorating circumstances.
"The situation is complicated and continues to worsen," blogger Yury Podolyaka said in a post on Telegram.
He said the town of Sudzha was "full of Ukrainian soldiers".
"Sudzha has been completely lost," he said.
"The enemy is entrenching, indicating that the fighting is likely to be long-term," the Dva Mayora Telegram channel said.
Multiple military bloggers reported Ukrainian troops had advanced towards the town of Korenevo, more than 25 kilometres from the Ukrainian border.
Russian energy giant Gazprom said on Thursday it would continue shipping gas through the Sudzha metering station, the last transit point for Russian gas heading to Europe via Ukraine.
Russia's defence ministry said its forces had destroyed a US-made Bradley Fighting Vehicle that Ukraine was using in the incursion and that it had downed Ukrainian drones over the region.
The governor of the Kursk region introduced a local state of emergency on Wednesday evening, a move that gives authorities the power to restrict movement in a bid to bring the situation under control.
Several thousand have been evacuated on both sides of the border.
