He said the Ukrainian side had received 1000 killed soldiers while Russia received 41 bodies.
The politician is responsible in the Russian parliament for matters relating to the "special military operation," as the war is officially called in Moscow.
The previous exchange of remains took place in February.
The office of prisoner-of-war affairs in Kyiv confirmed on Telegram that Ukraine had received 1000 bodies.
The authority thanked the International Committee of the Red Cross for its support.
Forensic experts would now examine the bodies, it said.
This is intended to rule out, among other things, that bodies of Russian soldiers were handed over by mistake.
The staff added that "after the identification of the deceased, the bodies will be handed over to the families for dignified burial".
Russia and Ukraine exchange soldiers' bodies on a regular basis.
Last year alone, Ukraine received back more than 15,000 bodies of fallen soldiers, according to official figures.
Russia received only a fraction of that.
Due to the slow but steady advance of Russian troops, the Ukrainian army is not always able to recover its own fallen soldiers.
Ukraine has been defending itself against a Russian invasion with help from allies for more than four years.
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