Trump advised Putin not to escalate the war in Ukraine and reminded him of "Washington's sizeable military presence in Europe", the Post reported.
During the election campaign, Trump said he would find a solution to end the war "within a day," but did not explain how he would do so.
Trump had also spoken to Ukrainian President Volodymyr Zelenskiy on Wednesday, according to media reports.
On Friday, the Kremlin said Putin was ready to discuss Ukraine with Trump but that did not mean that he was willing to alter Moscow's demands.
On June 14, Putin set out his terms for an end to the war: Ukraine would have to drop its NATO ambitions and withdraw all of its troops from all of the territory of four regions claimed by Russia.
Ukraine rejected that, saying it would be tantamount to capitulation, and Ukrainian President Volodymyr Zelenskiy has put forward a "victory plan" that includes requests for additional military support from the West.
(Reporting by Humeyra Pamuk and Brendan O'Brien; Editing by David Gregorio)

US President Donald J. Trump (L) and Russian President Vladimir Putin (R) hold a joint press conference in the Hall of State at Presidential Palace following their summit talks, in Helsinki, Finland, 16 July 2018. EPA-EFE/ANATOLY MALTSEV
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