Ukraine's top commander has said Kyiv's forces control 1,000 sq km of Russian territory as they press their biggest cross-border incursion in two-and-a-half years of full-scale war.

Commander Oleksandr Syrskyi said Ukraine continued to "conduct an offensive operation in the Kursk region" seven days after it began.

Ukraine's President Volodymyr Zelensky said Russia had brought war to others and now it was coming back to Russia.

But Russian leader Vladimir Putin described the offensive as a "major provocation" and ordered Russian forces to "to kick the enemy out of our territory".

“The only people who are in Ukraine are the Russians. They’re the ones that invaded Ukraine and Ukraine is defending itself against that aggression,” National Security Council spokesperson John Kirby said on Monday.

Kirby added that if Putin “doesn’t like it, if it’s making him a little uncomfortable, then there’s an easy solution. You can just get the hell out of Ukraine and call it a day.”