The "offer" of another summit came during a speech Kim gave to Parliament.
But, in an ominous note, Kim also touted his "treasured sword for defending the sovereignty of the country", an allusion to his nuclear weapons program.
But Kim said he did not feel the need to meet Trump just to secure sanctions relief, and said the prospect of another summit like February's meeting in Hanoi was "not inviting". But he said his personal relationship with the American president remains good.
Said Mr Kim Hyun-wook, a professor at the Korean National Diplomatic Academy: "They want to see the U.S. make the next move - that the steps that they took, such as returning war remains and shutting down missile sites, are even more grounds to demand the United States to relieve sanctions".
Asked about Kim's comments, media reports quoted South Korea's presidential office as saying Seoul is committed to keeping the atmosphere of dialogue alive and helping negotiations between Washington and Pyongyang resume at an early date.
Pyongyang will give the USA till the end of the year to drop "its current calculation method" with regards to bilateral talks and come up with "a correct posture" instead, Kim said, the state-run KCNA news agency reported on Saturday.
"In any case we will wait with patience for the United States courageous decision by the end of this year but it will clearly be tough to get such a good opportunity like the last time", Mr Kim said in a speech at North Korea's rubber-stamp legislature, according to the KCNA report.
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The next round of talks, held in Hanoi, Vietnam, quickly fell apart. Before he starts dismantling his missile and weapons programs, he wants the sanctions lifted and aid delivered to his country.
Analysts say it may also indicate a change in the country's constitution that concentrates more authority in the young leader. How long are we going to keep falling for this steaming pile of lies?
Born in 1950, Choe is considered Kim Jong Un's right-hand man - frequently referred to as the regime's "virtual number two official", having been key to the party's hierarchy since the 1980s. "Our government will do what we can in order to maintain the current momentum for dialog and help negotiations between the U.S. and North Korea resume at an early date", Mr Moon's office said in a text message. World leaders pay attention to him and treat him as if he's some sort of normal leader.
Van Jackson, a former Pentagon official who now teaches at Victoria University in New Zealand, said time was on North Korea's side.
Pyongyang has been careful not to criticise Trump personally, while attacking sanctions on North Korea as an attempt to "destroy modern civilisation and turn the society back in a medieval dark age".
The reshuffle comes at a time of heightened diplomatic activity that has seen Kim Jong Un hold landmark summits with the leaders of the US, China and South Korea.
North Korean media touted Kim's "re-election" as chair of the State Affairs Commission as an expression of "the general will of all the Korean people" and their gratitude for Kim's "outstanding ideological and theoretical wisdom and experienced and seasoned leadership and demonstrating the politico-ideological might and tremendous power of socialist Korea on the highest level".