MPutin gave new life to the prisoner exchange process on Thursday, when he told the eastern economic forum in Vladivostok that a "fairly large" exchange was close to the agreement.
"We are approaching the conclusion of the negotiations we are negotiating with the official authorities, among others," he said. "I think this will become known in the near future."
Ukrainian officials had said that the exchange of 33 prisoners on each side would take place last Friday, just to make Moscow reject it.
Among other things, Kiev hopes to bring home 24 captured sailors when their three navy ships were seized in November in an attempt to cross the Kerch Strait, which from the annexation of the Crimea was controlled by Russia.
In recent weeks, several Ukrainian prisoners in Russia have been transferred from distant prison colonies to Moscow.
The dramatic seizure of Mr. Tsemakh in the separatist territory by a team of Ukrainian agents who reportedly drugged him and smuggled him across the de facto wheelchair was the first of its kind.
About 13,000 people have been killed in the conflict in eastern Ukraine since 2014.