Russia President Vladimir Putin ordered his army on Monday to "dislodge" Ukrainian troops who entered Russian territory as authorities said over 120,000 people had been evacuated away from the fighting.

Kyiv launched a surprise offensive into Russia's western Kursk region Tuesday, capturing over two dozen settlements in the most significant cross-border attack on Russian soil since World War II.

Ukraine's military chief Oleksandr Syrsky told President Volodymyr Zelenskyy in a video posted Monday that his troops now control about 1,000 square kilometres of Russian territory and are continuing "offensive operations."

"One of the obvious goals of the enemy is to sow discord, strife, intimidate people, destroy the unity and cohesion of Russian society," Putin said in a televised meeting with government officials. "The main task is, of course, for the defence ministry to dislodge the enemy from our territories."

Some 121,000 people have fled the Kursk region since the start of the fighting, which has killed at least 12 civilians and injured 121, regional governor Alexei Smirnov said at the meeting with Putin.

Authorities in Kursk announced Monday they were widening their evacuation area to include Belovsky district, home to some 14,000 residents. The neighbouring Belgorod region also said it was evacuating its border district of Krasnoyaruzhsky.

Ukraine has pierced into the region by at least seven miles and has captured 28 towns and villages, with the new front 40 kilometres long, Smirnov said.

But Syrsky said that "as of now, about 1,000 square kilometres of Russian territory are under control," suggesting the area captured is more than twice as large.

He said that fighting was ongoing along almost the whole front and "the situation is under our control."

Zelenskyy said that he had instructed officials to "prepare a humanitarian plan for the area of the operation."

Putin said Russia would respond by showing "unanimous support for all those in distress" and claimed there had been an increase in men signing up to fight.

"The enemy will receive a worthy riposte," he said.

The assault appeared to catch the Kremlin off guard. Russia's army rushed in reserve troops, tanks, aviation, artillery and drones in a bid to quash it.

But it conceded on Sunday that Ukraine had penetrated up to 20 miles into Russian territory in places.

It said some forces were near the villages of Tolpino and Obshchy Kolodez, around 25 kilometres and 30 kilometres from the Russia-Ukraine border.

A Ukrainian security official told AFP, on condition of anonymity over the weekend, that "the aim is to stretch the positions of the enemy, to inflict maximum losses and to destabilise the situation in Russia as they are unable to protect their own border".

The Ukrainian official said thousands of Ukrainian troops were involved in the operation.

Russia's defence ministry said on Monday that its air defence systems had destroyed 18 Ukrainian drones, including 11 over the Kursk region.