The Mariupol City Council had said earlier that the evacuation of about 400,000 people besieged by Russian forces in the cities of Mariupol and Volnovach would begin at noon at 1000 GMT.
Officials were forced to abandon a similar plan on Saturday after there were some violations of the ceasefire and the two sides blamed each other for it.
On Saturday, Russia and Ukraine accused each other of failing to provide safe passage to civilians fleeing the two cities, amid a war that has caused Europe’s biggest humanitarian crisis in decades.
The war, which began with a major Russian offensive on February 24, led to the flight of nearly 1.5 million refugees west to the European Union, and resulted in unprecedented international sanctions on Moscow, as well as a warning of a global economic recession.
The Russian Defense Ministry said units of them had opened humanitarian corridors near Mariupol and Volnovakha, which are cordoned off by Russian forces.
But the city council in Mariupol said that Russia was not implementing the ceasefire agreement needed to provide safe passage, and that it had therefore asked residents to return to shelters and wait for new information on the evacuation.
On the other hand, the Russian Defense Ministry accused what it calls “Ukrainian nationalists” of preventing civilians from leaving the two cities.
Mariupol, a port in southeastern Ukraine, was hit by heavy bombardment, indicating the strategic importance Moscow attaches to it because of its location between Russian-backed separatist-held areas in eastern Ukraine and the Black Sea peninsula of Crimea, which Russia annexed from Ukraine in 2014.
So far, there is no electricity, water, heating or cell phone connections in the city, and food supplies are running out.
The Ukrainian government said that its plan is to evacuate 200,000 residents of Mariupol and 150,000 from Volnovakha.
The Russian “TASS” news agency quoted the Moscow-backed militants as saying that only 17 people left Mariupol on Saturday, while no one left Volnovakha.
