The swap on Friday was not yet finished, according to the official, who spoke on condition of anonymity because he was not authorised to speak publicly.
Moscow did not immediately confirm the exchange was under way.
Ukraine and Russia agreed to the exchange of 1000 prisoners from each side a week ago in Turkey in their first direct peace talks since the early weeks of Moscow's 2022 invasion of its neighbour.
That meeting lasted only two hours and brought no breakthrough in international diplomatic efforts to stop the fighting.
The Ukrainian comment Friday came after US President Donald Trump said Russia and Ukraine had carried out a large exchange of prisoners.
"A major prisoners swap was just completed between Russia and Ukraine," Trump said on the Truth Social platform.
He said it would "go into effect shortly", although it was not clear what that meant.
"This could lead to something big???" Trump added in his post, apparently referring to international diplomatic efforts to stop the fighting.
White House and National Security Council officials did not immediately respond to requests for further details.
After the May 16 talks, Turkish Foreign Minister Hakan Fidan called the prisoner swap a "confidence-building measure" and said the parties had agreed in principle to meet again.
But Kremlin spokesman Dmitry Peskov said Friday that there has been no agreement yet on the venue for the next round of talks as diplomatic manoeuvring continued.
European leaders have accused Russian President Vladimir Putin of dragging his feet in peace efforts while he tries to press his larger army's battlefield initiative and capture more Ukrainian land.
The Istanbul meeting revealed both sides clearly remained far apart on key conditions for ending the fighting.
One such condition for Ukraine, backed by its Western allies, is a temporary ceasefire as a first step toward a peaceful settlement.
The Kremlin has pushed back on a temporary halt to hostilities, and Putin has said any such truce must come with a freeze on Western arms supplies to Ukraine and an end to Ukraine's mobilisation drive.
A senior Ukrainian official said that in Istanbul, Russia had introduced new, "unacceptable demands" to withdraw Ukrainian forces from huge swaths of territory.
The official, who was not authorised to make official statements, spoke to the AP on condition of anonymity.
The proposal had not been previously discussed, the official said.
Putin has long demanded that Ukraine withdraw its troops from the four regions that Russia annexed in September 2022 but never fully occupied as a key condition for a peace deal.
Ukrainian President Volodymyr Zelenskiy has warned that if Russia continues to reject a ceasefire and make "unrealistic demands", it will signal deliberate efforts to prolong the war - a move that should bring tougher international sanctions.
