Moscow is pressing its offensive to retake the full territory of Russia’s Kursk region from Ukraine as negotiations between the White House and the Kremlin continue over a possible cease-fire in the three-year war.

Russia’s Defense Ministry said on Saturday that its forces had retaken two villages outside Sudzha, the main Russian town that Ukraine occupied since its surprise offensive into Russia last summer, but that it appears to have lost in recent days. Ukrainian officials did not immediately comment on Russia’s newest claim and have not confirmed a retreat by its forces from Sudzha.

Moscow’s apparent advances on the Kursk front came a day after President Vladimir V. Putin of Russia called on Ukrainian forces still fighting in the region to lay down their arms. Mr. Putin said he would spare their lives if they surrendered.

The Russian leader has said that Ukrainian forces are encircled in the region, an assertion that President Trump repeated in a message on Truth Social. But Ukraine’s military said on Friday that “there is no threat of encirclement of our units,” calling the suggestion “false and fabricated by the Russians.”

The Kremlin spokesman, Dmitri S. Peskov, reiterated Mr. Putin’s demand in comments to the state news agency Tass on Saturday.

“It’s still valid,” Mr. Peskov said, although he added that “time was running out.”

Russia and Ukraine have both been ramping up their attacks. Overnight, Russia launched two Iskander-M ballistic missiles and 178 drones at Ukrainian cities, the Ukrainian authorities said. The missiles struck a residential district in Kryvyi Rih, President Volodymyr Zelensky’s hometown in central Ukraine, injuring 14 people, including two children, according to the local authorities.