Combined images of (From L to R) US President Donald Trump and Russian President Vladimir Putin.
Trump, who has been pressing Russia to agree on a 30-day unconditional ceasefire, said he would speak with him by phone on Monday. Kremlin spokesman Dmitri Peskov told the state TASS news agency the call was "being prepared".
Earlier Saturday, the Kremlin had said that a meeting between Putin and Ukrainian President Volodymyr Zelensky would be possible only after both sides reach an agreement.
That came a day after the first direct talks between the two countries in more than three years led to an agreement for another exchange of prisoners.
Early Saturday, a Russian drone attack on a minibus carrying evacuated civilians in Ukraine's eastern Sumy region killed nine people and wounded five, local authorities said.
Zelensky, denouncing the attack and Russia's refusal so far to agree on a ceasefire, repeated his call for reinforced sanctions against Moscow.
"Without stronger sanctions, without stronger pressure on Russia, there will be no real diplomacy there," he insisted.
On Friday in Istanbul, the first direct Ukraine-Russia talks since the spring of 2022 -- shortly after Moscow's full-scale invasion that February -- led to an agreement to exchange 1,000 prisoners each.
Ukraine's top negotiator, Defence Minister Rustem Umerov, said the "next step" would be a meeting between Zelensky and Putin.
Russia said it had taken note of the request.
"We consider it possible, but only as a result of the work and upon achieving certain results in the form of an agreement between the two sides," the Kremlin's spokesman said.
Trump denounces 'bloodbath'
Russia's top negotiator, Vladimir Medinsky, said that Moscow and Kyiv would "present their vision of a possible future ceasefire", without saying when.
The Kremlin said that first, the POW swap had to be completed, and both sides needed to present their visions for a ceasefire before fixing the next round of talks.
"For now, we need to do what the delegations agreed on yesterday", in Turkey, Kremlin spokesman Dmitry Peskov said, which meant "first and foremost to complete a 1,000 for 1,000 swap".
The head of Ukraine's military intelligence, Kirillo Budanov, told broadcaster TSN he hoped the exchange would happen next week.
Posting on Truth Social Saturday, Trump said he would speak to Putin on Monday to discuss finding a way out of the "BLOODBATH".
Afterwards, he added, he would speak to Zelensky and NATO officials, expressing hope that a "ceasefire will take place, and this very violent war... will end".
Both Moscow and Washington have already talked up the need for a meeting on the conflict between Putin and Trump.
Trump has argued that "nothing's going to happen" on the conflict until he meets Putin face-to-face.
US Secretary of State Marco Rubio welcomed the latest prisoner exchange in a telephone call with Russian Foreign Minister Sergei Lavrov, his spokeswoman Tammy Bruce said.
But he underlined Trump's call "for an immediate ceasefire and an end to the violence", she added.
Short link: