MARCH 30: Mr Putin made the comments during an hour long phone call with French President Emmanuel Macron on Tuesday night, the Kremlin said in a statement.
But French officials said the Russian leader had agreed to consider plans to evacuate civilians from the city.
It comes as new satellite photos showed the destruction caused by the shelling.
The images, released by the Earth observation company Maxar, showed that residential areas have been reduced to rubble and highlighted Russian artillery cannons in firing positions on the outskirts of the city.
Officials from France's Elyseé palace called the situation in the city "catastrophic" and added that "civilian populations must be protected and must leave the city if they wish to. They must have access to food aid, water and the medicines they need".
"This very degraded humanitarian situation is linked to the siege of the city by the Russian armed forces," the statement said.
France, along with Turkey, Greece and several humanitarian groups, have presented Mr Putin with a plan to evacuate the city.
Officials said that Mr Putin told Mr Macron that he will "think about" the proposal.
But in its readout of the call, the Kremlin appeared to suggest that Mr Putin has provided no such assurances.
Russian officials said Mr Putin told the French leader that "in order to resolve the difficult humanitarian situation in this city, Ukrainian nationalist militants must stop resisting and lay down their arms".
The statement added that Mr Putin had given Mr Macron "detailed information about measures taken by the Russian military to provide emergency humanitarian assistance and ensure the safe evacuation" of civilians from the besieged south-eastern city.
Ukraine has accused Russia of forcibly relocating thousands of people from Mariupol to Russian-held territory.
The call followed claims by the mayor of Mariupol that thousands of people have been killed during the Russian bombardment of the port city.
Vadym Boychenko, who has been evacuated from the city, told the Reuters news agency that almost 5,000 people, including around 210 children, have been killed since Russian shelling began.
Matilda Bogner, the head of the UN human rights mission in Ukraine, told Reuters that she believes "there could be thousands of deaths, civilian casualties, in Mariupol".