The governor of Kursk said on the messaging service Telegram that a state of emergency was declared in the region due to the attacks.
The United States says it is discussing with Ukraine an attack on Russian territory.
According to Russia, Ukraine launched a large-scale attack on the Kursk region on the Russian border on Tuesday.
White House Press Secretary Karine Jean-Pierre said the United States does not have complete information about the attack.
Ukrainian representatives have so far not commented on the attack.
The governor of Kursk said on the messaging service Telegram that a state of emergency was declared in the region due to the attacks.
Sources: AFP, Reuters