Putin congratulated Kim on Friday on his re-election as head of the North Korean State Affairs Commission and confirmed his readiness to cooperate on pressing regional matters.
Moon also said he won't be restricted by location or format to have his fourth summit with Kim, indicating that he will not hold Kim to a promise to meet next in Seoul.
Moon said he hoped to make concrete and effective discussions with Kim "in any place and formality at the North's convenience".
Moon's office did not provide a timeline on when his fourth summit with Kim might take place. "I look forward to the day, which could be soon, when Nuclear Weapons and Sanctions can be removed, and then watching North Korea become one of the most successful nations of the World!".
However, Kim said he remained committed to improving relations with South Korea if it showed its "sincerity by practical action, not by words". He also lobbied hard to set up the first summit between Kim and Trump last June, when they agreed to a vague statement about a nuclear-free Korean Peninsula without describing how or when it would occur.
"Anyway, we will wait for a bold decision from the USA with patience till the end of this year, but I think it will definitely be hard to get such a good opportunity as the previous summit", he added, underscoring a deadline of sorts for the Trump administration. After the meeting with Moon, Trump said that "smaller deals" could happen "step by step" between the north and the USA, though his country and South Korea were focusing on "the big deal".
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Experts said North Korea is trying to increase its leverage in future dealings with Washington by tightening cooperation with Moscow while requesting more humanitarian support. South Korea: Why has the peninsula been divided?
"He also expressed his willingness for the resumption of dialogue between North Korea and the United States and a third summit between North Korea and the United States", Moon said about Kim.
At their February 27-28 summit in Hanoi, the USA and North Korean leaders fell short of bridging the gap between Washington's denuclearization demands and Pyongyang's calls for sanctions relief.
In a speech to Pyongyang's rubber-stamp parliament on Friday, Kim said the Hanoi meeting raised questions about Washington's intention but added he will wait until the end of the year for the United States to make "a courageous decision".
Kim blamed the collapse of the second summit with Trump on what he described as Washington's unilateral demands. He said the North's economy would preserve through heavy USA -led sanctions imposed over its nuclear weapons program and that he "wouldn't "obsess over summitry with the United States out of thirst for sanctions relief".
South Korean President Moon Jae-in speaks during a meeting with his aids at the presidential Blue House in Seoul, South Korea, Monday, April 15, 2019.
But Moon did not specifically respond to Kim's criticism that South Korea was too subservient to the "anachronistic arrogance and hostile policy of the United States", and that the South's military persisted in "veiled hostility" by conducting exercises with USA forces.