The Ukrainian army has taken control of several communities since it launched its surprise offensive in the Russian region of Kursk on August 6.
"We are continuing to advance in the Kursk region. Between one and two kilometers in several different areas during the day," says Zelenskyj in a statement via social media.
Ukraine intends to establish a "buffer zone" in the region to prevent Russian attacks across the border, says Interior Minister Ihor Klymenko.
Russia claims simultaneously that it has repelled Ukrainian advance attempts in five areas in Kursk.
Ukrainian forces have now advanced over an area of at least 800 square kilometers, according to analyses of data from the American think tank ISW. Over 120,000 people have fled their homes in Kursk since the ground offensive began.
In Ukraine, millions of people have been forced to flee from Russian forces since the Kremlin launched its full-scale invasion of the country nearly two and a half years ago.