Ushakov said Trump made the offer during a 90 minute phone call on Saturday, US Independence Day, in the context of his participation next week at the NATO summit in Turkey.
Ukrainian President Volodymyr Zelenskiy said he had also spoken to Trump.
"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 of Trump's call with the Russian president.
Ushakov, who described the conversation as "business-like and quite constructive," said Russia sought "a political-diplomatic resolution of the conflict, with due account of Russia's fundamental approach."
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, which have triggered fuel shortages in several Russian regions.
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's Donetsk region.
Zelenskiy and Ukraine's general staff dismissed that claim on Saturday, saying Kyiv's forces still controlled the city.
"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 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.
Russia has said any solution must include Moscow assuming full control over Ukraine's Donbas region.
Ukraine rejects that assertion and Zelenskiy urged Putin last month to hold a one-on-one meeting with him, but the Kremlin leader refused.
Ushakov 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.
US diplomatic efforts have virtually stalled as Washington is focused on the war with Iran.
Ushakov said Putin expressed hope during the conversation that US diplomatic efforts in the Iran conflict would "allow for mutually acceptable long-term solutions to be found on key issues of a settlement".
Ushakov said Putin also reminded Trump that he had an open invitation to visit Moscow.
Zelenskiy, writing on his Telegram account, described his conversation with the US president as "very good," including a discussion on the war's 1200-km front line.
"There is a real prospect to end this war and American resolve will have a crucial meaning," he said. Zelenskiy said he and Trump agreed to continue discussions at the NATO meeting.
with DPA
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