Russia's defence ministry said its air defence units destroyed 12 drones over the Kursk region, which Ukraine attacked a week ago and claims to have captured 1,000 square kilometres (386 square miles) by Monday.
Putin said on Monday that Ukraine's incursion - the largest since he launched Moscow's full-scale invasion of Ukraine in 2022 - was aimed at improving Kyiv's negotiating position ahead of possible peace talks.
In his most detailed public remarks on the attacks to date, he also said that Kyiv "with the help of its Western masters" is trying to slow down the advance of Russian forces in what the Kremlin calls a "special military operation" in Ukraine.
He also questioned what negotiations there could be with an enemy he accused of firing indiscriminately at Russian civilians and nuclear facilities.
On Monday, Ukraine's top commander Oleksandr Syrskyi told President Volodymyr Zelenskiy on Monday troops were continuing their advance into Russian territory.
"We continue to conduct an offensive operation in the Kursk region. Currently, we control about 1,000 square kilometres of the territory of the Russian Federation," he said in a video published on Zelenskiy's Telegram account.
Reuters was unable to verify the claims by either side.
The acting governor of Kursk, Alexei Smirnov, said Ukraine controlled 28 settlements in the region, and the incursion was about 12 km (7.5 miles) deep and 40 km (25 miles) wide.
In Kursk, 121,000 people had already left or have been evacuated and another 59,000 were in the process of being evacuated, local officials said. In Russia's Belgorod region, which borders Kursk, 11,000 civilians were also evacuated, the region's governor said.
Ukrainian forces in Kursk were trying to encircle Sudzha, where Russian natural gas flows into Ukraine, while major battles were under way near Korenevo, about 22 km (14 miles) from the border, and Martynovka, according to some military bloggers.
Ukraine's air force said on Tuesday that Russia launched 38 attack drones and two ballistic missiles on Ukraine overnight. The whole country was briefly under air raid alerts early on Tuesday, but these have since been called off.
