But Pompeo said that in his talks with Lavrov, he "underscored that we must maintain full implementation of the United Nations sanctions until the final, fully verified denuclearization of North Korea is achieved".
Pompeo said improved ties between the two countries - damaged by Russia's annexation of Crimea in 2014, Russian interference in the 2016 US presidential election and USA sanctions against Russia - "will be good not only for our two countries but for the whole world as well".
And he did say that the US does not want to have war.
After the meeting, Kremlin aide Yuri Ushakov said Pompeo told Putin that Trump is ready to hold talks with the Russian leader during the G20 summit in Japan's Osaka in June.
Pompeo was the highest-ranking American to see Putin since July when the Russian leader met in Helsinki with Trump - who shocked the USA establishment by seeming to take at face value Putin's denials of election interference.
The exchange follows the publication of the Mueller report, which found that Russian Federation interfered in the 2016 USA election but did not conclude that the Trump campaign colluded with Moscow.
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Earlier, Pompeo said he warned Russian Federation against meddling in the 2020 USA presidential elections.
"We highlighted very useful cooperation that's happening on Afghanistan, including the trio format", Lavrov said alongside Pompeo on Tuesday.
Lavrov told reporters the two countries continued to disagree about Iran's nuclear program and the worldwide community's approach to Tehran, while Pompeo said he had raised USA concerns about escalating fighting in Syria's Idlib.
The U.S. has been critical of Russian Federation for propping up Maduro's regime.
"Most of all we must make sure we don't end up putting Iran back on the path to re-nuclearisation, because if Iran becomes a nuclear power its neighbours are likely to want to become nuclear powers", Hunt said.
Ties between the two countries have been poisoned by allegations - denied by Moscow - that Russian Federation tried to influence the results of the 2016 US presidential election, and by differences over Venezuela, Iran, Syria and Ukraine. "I hope that the necessary conditions for that to happen will now be created".
Speaking to reporters during a post-summit press conference in Vladivostok on April 25, Putin repeatedly suggested the main obstacle to progress in talks with the US remains the North's continued concerns over its security.
"It's not about 'moving on.' It's about trying to find solutions, compromises, places where there are overlapping interests", he told reporters. At the time, Trump cited Russia's failure to release Ukrainian ships and sailors seized during clashes in the Kerch Strait near Crimea as the reason for his refusal to meet Putin.