The Russian defence ministry on Saturday announced it had cleared the entire urban area of Mariupol of Ukrainian forces and said only a few fighters remained in the Azovstal steelworks, the scene of repeated clashes.
In an online post, the ministry said that as of April 16, Ukrainian forces in the besieged port city had lost more than 4,000 people, RIA added.
Russian forces have been trying for several weeks to take the Black Sea port.
"The entire urban area of ??Mariupol has been completely cleared ... remnants of the Ukrainian group are currently completely blockaded on the territory of the Azovstal metallurgical plant," the ministry said.
"Their only chance to save their lives is to voluntarily lay down their arms and surrender."
There was no immediate reaction from Kyiv to the statement by the Russian ministry, which also said 1,464 Ukrainian servicemen had surrendered so far.
Moscow said the total number of what it called "irretrievable losses" suffered by Ukraine totaled 23,367 people but did not provide any evidence and did not say whether this included only those who had died or who had also been injured.