A convoy of 25 buses, ambulances and aid was due in the Russian-occupied city on Friday, delivering food and medicine and evacuating people who want to leave to Russia and Ukraine.
Some 2000 people are still living there, authorities and volunteers say.
A localised ceasefire in Ukraine's southern Kherson region has held for four days and is due to continue through to Sunday evening.
"Everyone who wishes to leave Oleshky will have the opportunity to do so safely," the warring countries said in identical statements.
"Those who choose to stay may remain and will receive all the humanitarian support they need."
Officials from Russia and Ukraine worked together on the operation in a rare joint humanitarian effort.
The United States played a central role in the mission, with the White House saying it was brokered by first lady Melania Trump.
Pope Leo XIV also pressed the Russian and Ukrainian leaders to help Oleshky.
But outside Oleshky and its surroundings the war continued.
Russia's defence ministry said that air defences downed more than 1000 Ukrainian drones overnight, and Ukraine's air force said Russia launched ballistic and other missiles as well as scores of drones.
The convoy was set up in Armiansk, a town in Russian-occupied southern Ukraine about 85km south of Oleshky, said Dmytro Lubinets, Ukraine's ombudsman and chief negotiator.
Demining of designated routes into the city was completed late on Thursday, officials said, adding that two people were killed during the work.
Lubinets said the evacuation would run until 2pm, local time, and that aid would be begin being distributed 30 minutes later.
It was not clear if he was using Ukrainian time or western Russian time, which is an hour ahead.
The route that evacuees would take was not published.
Ukrainian territory is just across the River Dnipro from Oleshky, a couple of kilometres away.
A volunteer from Oleshky, Ksenia Arkhipova, told The Associated Press on Thursday that roughly 1000 residents had told her they want to leave the city.
She said it was important that the Ukrainian delegation was present in Oleshky because many residents feared Russian authorities would interrogate or detain them.
Authorities have invited a small number of Russian, Ukrainian and US media organisations to cover the operation in Oleshky.
