Ukrainian President Volodymyr Zelenskiy, who was on a phone call with Trump on Wednesday, had earlier urged the United States to monitor a ceasefire between Ukraine and Russia.
Putin had agreed to pause attacks on energy infrastructure, a proposal Zelenskiy accepted despite denouncing Russia for rejecting a more comprehensive truce.
Even that narrow pause appeared in doubt on Wednesday, however, with Russia saying Ukraine hit an oil depot in southern Russia while Ukraine said Russia had struck hospitals and homes and also knocked out power to some of its railways.
Still, the two sides announced they had carried out a swap of prisoners, each releasing 175 troops in a deal facilitated by the United Arab Emirates.
Russia said it also freed an additional 22 wounded Ukrainians as a goodwill gesture.
Zelenskiy said on Wednesday that the continued attacks showed Russia's words did not match its actions, and that Russia was not ready for peace.
He said the United States should be put in charge of monitoring any ceasefire.
"If the Russians will not strike our facilities then we will definitely not strike theirs," he said at a briefing in Helsinki alongside the president of Finland.
The Kremlin said it had called off planned attacks on Ukraine's energy infrastructure, including by shooting down seven of Russia's own drones heading towards Ukraine.
It accused Ukraine of failing to call off its own attacks in what it called an attempt to sabotage the agreement.
