Most people trapped in the port city are sleeping in bomb shelters to escape more than six days of near-constant shelling by encircling Russian forces that has cut off food, water, power and heating supplies, according to the Ukrainian authorities.
The civilian death toll from hostilities across Ukraine since Moscow launched its invasion on 24 February stood at 364, including more than 20 children, according to the United Nations on Sunday, with hundreds more injured.
The UN High Commissioner for Human Rights said most civilian casualties were caused by the use of "explosive weapons with a wide impact area, including shelling from heavy artillery and multi-launch rocket systems, and missile and air strikes."
Moscow has repeatedly denied attacking civilian areas.
In Irpin, a town some 25km northwest of the capital Kyiv, men, women and children trying to escape armed clashes in the area were forced to take cover when missiles struck nearby, according to Reuters witnesses.
Soldiers and fellow residents helped the elderly hurry to a bus filled with frightened people, some cowering as they waited to be driven to safety.
The invasion has drawn almost universal condemnation around the world, sent more than 1.5 million Ukrainians fleeing from the country, and triggered tough Western sanctions against Russia aimed at squeezing its economy.
Putin made his demand for Kyiv to end the fighting in a phone call with Turkish President Tayyip Erdogan, who appealed for a ceasefire. Putin told Erdogan he was ready for dialogue with Ukraine and foreign partners but any attempt to draw out negotiation would fail, a Kremlin statement said.
Russian media said Putin also held almost two hours of talks on Sunday with French President Emmanuel Macron, who has stayed in regular contact.