Kremlin aide Yuri Ushakov said the superpower leaders spoke by telephone for nearly 90 minutes, in comments made public early on Sunday.
Ushakov said Trump made the offer in the context of his participation next week at the NATO summit in Turkey.
"The American president once again confirmed his readiness to work towards a rapid end to the fighting and find solutions to overcome the crisis," Ushakov said.
Russia sought "a political-diplomatic resolution of the conflict, with due account of Russia's fundamental approach", he said.
Ushakov accused Kyiv and its European allies of "counting on extending and even escalating the conflict, and on terrorism against civilians".
He was referring to Ukraine's long-range strikes on Russian targets, mainly linked to the oil industry.
He quoted Trump as saying that Washington's envoys, Steve Witkoff and Jared Kushner, would keep trying to broker a settlement and were prepared to make another visit to Moscow.
Ushakov said Putin "depicted the real situation on the battlefield where the Russian armed forces are confidently advancing, liberating one locality after another".
Russian commanders told Putin on Friday that Moscow's troops had captured the strategically important city of Kostiantynivka in eastern Ukraine.
On Saturday, President Volodymyr Zelenskiy and Ukraine's General Staff rejected that claim, saying Kyiv's forces still controlled the city.
Russia's military ignored Zelenskiy's assertions and said it was mopping up remaining pockets of resistance and offered to exchange the remains of dead servicemen.
"Of course, that is not true. It is just another Russian lie, an attempt to generate some kind of a news story," Zelenskiy said on X.
"If Kostiantynivka were under Russian control, then perhaps (Russian President Vladimir) Putin would have no problem meeting me there to find a diplomatic way to finally end this war," he added.
The General Staff also said Kostiantynivka remained under the control of Ukrainian forces.
"Military units and subunits of the 19th Army Corps of the Eastern Grouping continue to conduct defensive operations on designated lines within the town and on its approaches," it said in a statement.
A Russian defence ministry statement said: "With the liberation of the city of Kostiantynivka by Russian troops and the establishment of full control over it, the Russian side is ready to carry out a humanitarian action to hand over the bodies of deceased Ukrainian servicemen in this settlement."
The statement on Telegram said Russia was proposing a six-hour ceasefire on Monday around the town to permit an exchange and gave Ukraine until noon on Sunday to reply.
Russian state news agency TASS quoted a commander in the town, identified by call sign Agalu, as saying that clearing operations in the town were "coming to an end" and Russian troops were already setting up battle positions.
Kostiantynivka, which once had a population of 67,000, is the southernmost of four key settlements that form a defensive line central to Ukraine's effort to hold the heavily industrialised Donetsk region.
Analysts say capturing the city would give Russian forces a foothold from which to push north along the defensive belt, now the main axis of their campaign.
Russia's military has for some time reported controlling parts of Kostiantynivka, one of several heavily fortified cities that make up Ukraine's so-called "fortress belt" in Donetsk.
Aside from Kostiantynivka, Russia's defence ministry quoted another commander as saying heavy fighting was continuing around Lyman, a town further northeast.
The ministry also said its forces had taken five villages in eastern Ukraine: Shyikivka, Novyi Myr, Cherneshchyna and Druzhelyubivka in Kharkiv region and Vasylivka in Donetsk region.
Reuters could not verify the battlefield reports.
with DPA
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