Ukraine's general staff admits in a statement that "the operational situation is difficult" but adds that "our units are holding their positions" in Kursk.
Earlier on Saturday, news came from the Kremlin that Russian Army Chief Valerij Gerasimov had informed President Vladimir Putin that they had been "completely liberated" from Ukrainian control.
Putin congratulated the Russian soldiers in a statement, where he also claims that Kyiv's "adventure in Kursk has completely failed".
The enemy's total defeat in the border region of Kursk creates the right conditions for further successes for our forces and at important front sections, said Putin.
Gerasimov also confirmed that North Korean soldiers had participated in the battles and says that they had provided "significant help in defeating the Ukrainian forces".
North Korea has sent over 10,000 soldiers to fight on Russia's side, according to South Korean and Ukrainian reports.
Moscow's triumphant statement comes on the same day as US President Donald Trump met with Ukraine's President Volodymyr Zelenskyj in the Vatican, in connection with Pope Francis' funeral. Trump has recently stated that Russia and Ukraine are "very close" to an agreement that can lead to peace.
The Ukrainian forces have held control over parts of the Russian Kursk region since a surprise offensive in August. In March, Vladimir Putin ordered the Russian military to "completely liberate" Kursk.
Russia launched its full-scale invasion war in Ukraine over three years ago.