Trump's shock election victory has been met with trepidation in Kiev because of the outspoken reality TV star's praise of Russian President Vladimir Putin, and his apparent indifference to the Western coalition against Moscow.
The billionaire businessman suggested earlier this year the US could accept Russia's annexation of Crimea if it led to improved relations between the two nations, which are bitterly at odds over Syria.
Poroshenko congratulated Trump on his victory and said he wished "to work together with his administration to further strengthen the strategic partnership between Ukraine and the United States", according to a statement by the Ukrainian presidency.
The two men agreed to organise "a bilateral meeting", the statement said, without giving further details.
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The US election outcome had sparked fears in the ex-Soviet republic after Trump was accused several times by his Democratic presidential rival Hillary Clinton of being Putin's "puppet".
Earlier this week, Poroshenko said he had "no doubt" that Trump would refuse to recognise Russia's annexation of Crimea.
Putin and Trump spoke on the phone on Monday evening for the first time since the US vote, agreeing on the need to normalise ties between Washington and Moscow, the Kremlin said.
Russia also has been accused by Washington of supporting pro-Russian rebels in eastern Ukraine, which the Kremlin denies.