In their first summit last June in Singapore, Trump and Kim issued a vague statement calling for a nuclear-free Korean Peninsula without describing when or how it would occur.
The comment came after Moon visited Washington last week for talks with US President Donald Trump.
"Now, it's time to actively prepare and push for a South-North summit", Moon told a meeting of his senior secretaries, referring to a potential fourth summit between the two Koreas.
Moon met Kim three times a year ago and also brokered nuclear talks between North Korea and the US following tensions created by the North's nuclear and missile tests and the exchange of war threats by Kim and President Donald Trump. "S. summit with a certain methodology that can be shared with us, we can think of holding one more talk", Kim said.
On Thursday, Trump said sanctions imposed against North Korea will remain in place until Pyongyang fully dismantles its nuclear program.
A summit between U.S. President Donald Trump and Kim in Hanoi ended early without significant progress. "But U.S. -style dialogue of unilaterally pushing its demands doesn't fit us, and we have no interest in it".
"If the USA adopts a correct posture and comes forward for the third DPRK-U".
Facing all this, we can expect that the North Korean ruler is becoming frustrated with the negotiating process. South Korea: Why has the peninsula been divided? Some 28,500 US troops are stationed in the South.
As North Korea's economy is believed to have been hit hard by global economic sanctions aimed at preventing the country from developing nuclear weapons and ballistic missiles, many political slogans in Pyongyang have promoted "self-reliance".
Other state media outlets also published a series of messages sent from foreign organizations to celebrate Kim's birthday in an apparent effort to praise the late founder as a world-class leader. Moon's opening remarks made no mention of a special envoy. They will do so with the calculated expectation that the United States will rush to placate Pyongyang and offer concessions. "[North Korea] dialogues", Moon said, according to an English transcript from his office. Moon, instead, said his administration is determined to implement inter-Korean agreements step by step, even if there are obstacles. Kim was expected to visit Moscow before the end of a year ago, but it did not materialize.
That didn't happen, but the two sides did begin mine clearing operations in the so-called Demilitarized Zone and a railway survey in the North.
"South Korea has limits because it can't lift sanctions against North Korea so it basically has no leverage", Jeung said.