Kim said he will wait until the end of this year for the U.S. to decide if it wants another summit.
Kim used a speech in front of the Supreme People's Assembly on Friday to criticise the United States president, Donald Trump, and to question whether North Korea would enter into a third round of talks with America, following summits between the two nations in Hanoi and Singapore.
South Korea's Blue House said in a statement that officials would "do what we can in order to maintain the current momentum for dialogue and help negotiations between the USA and North Korea resume at an early date".
If the US offers a plan by the end of 2019 that both countries can accept, North Korea's leader can again meet with the USA president to continue diplomatic efforts towards peace, official North Korean news agency KCNA reported on Saturday.
Trump also says the main goal is to get rid of all North Korean nuclear weapons, and that once that happens North Korea has "potential as great as anything".
Kim has signed vague statements calling for the "complete denuclearisation" of the peninsula in his meetings with Trump and Moon. "But US-style dialogue of unilaterally pushing its demands doesn't fit us, and we have no interest in it", he added, referring to his second meeting with Trump in Vietnam earlier this year.
Korean state media also graced Kim with a new title, "Supreme Representative of All the Korean People", which is not mere hyperbole but an actual title ratified by government decree several months ago.
But the lack of substantial disarmament commitments from the North and the deepening impasse in nuclear negotiations have fueled doubts over whether Kim would ever voluntarily relinquish an arsenal he may see as his strongest guarantee of survival.
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Responding to Mr Kim's speech, South Korea repeated its stance of promoting the talks.
Speaking alongside South Korean President Moon Jae-in, Trump on Thursday expressed openness to "smaller deals" with the North.
Long-serving officials such as 91-year-old Kim Jong-nam, the titular head of state, and Premier Pak Pong-ju, 80, were either retired or promoted to symbolic posts and replaced by younger officials.
Analysts said, however, the existing strictures on Choe will have little impact on his future diplomatic endeavours, as long as its nuclear dialogue with the USA continues.
Kim made the comments during a speech Friday at a session of North Korea's rubber-stamp parliament, which made a slew of personnel changes that bolstered his diplomatic lineup amid stalemated negotiations with the United States.
Moon, for his part, has called for an easing of sanctions, including those holding back joint economic projects between North and South Korea.
Trump walked away from making a deal with Kim at their meeting in late February. But Trump said he is still looking for a wide-ranging agreement under which North Korea commits to completely dismantling its nuclear weapons.
A senior administration official said Thursday that the US and the worldwide community had "clearly defined the scope" of North Korea's weapons of mass destruction and ballistic missile programs, have a shared understanding of what final, fully verified denuclearization entails and what meaningful progress toward that goal looks like.