The visit comes after a summer of intense fighting, with Moscow advancing fast in eastern Ukraine and Kyiv holding on to swaths of Russia’s Kursk region.
It also comes as Kyiv has for weeks pressed the West to allow it to use long-range weapons to strike targets deep inside Russia – so far to no avail.
When they meet at the White House on Thursday, Mr Zelensky is expected to try to convince Mr Biden to change his mind.
Mr Zelensky said the coming weeks would decide how more than 30 months of fighting that has killed thousands would end.
“It is now being determined what the legacy of the current generation of states leaders will be. Those in the highest offices,” he said.