The remarks were clearly aimed at Washington, and they come weeks after a second summit between Trump and Kim in Hanoi that ended abruptly with no progress toward the USA goal of ending North Korea's nuclear weapons and ballistic missile programs.
The US President added that a "third Summit would be good in that we fully understand where we each stand".
But Washington insisted on what officials described as "a big deal" under which sanctions relief would be traded for the dismantlement of all of the North's nuclear and other weapons programs. "But US-style dialogue of unilaterally pushing its demands doesn't fit us, and we have no interest in it", Kim said during the speech.
Kim said he will wait until the end of this year for the United States to decide if it wants another summit.
Mr Kim's remarks suggest an attempt to breathe new life into the stalled negotiations.
"As President Trump has repeatedly mentioned, personal relations between me and President Trump are not as hostile as relations between the two countries, and we still maintain excellent relations and can exchange letters whenever we would like to", Kim said in comments reported Saturday.
Kim repeated earlier claims that North Korea's crippled economy would persevere through heavy global 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".
"However, the U.S. has taken a more hard-line stance since the Hanoi summit".
Moon and Trump had also met in Washington on Thursday, where both made conciliatory statements toward North Korea.
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"We were informed by the prison authorities that she would be released on May 3, and it is likely she will be flown back to Hanoi on the same date", he said, when contacted.
Kim Yong Nam, the now-91-year-old who held the position of the North's head of state - an nearly entirely ceremonial role - for nearly 20 years, was replaced by Choe Ryong Hae, the state-run KCNA said.
South Korean President Moon Jae-in met Trump in Washington on Thursday in a bid to keep the dialogue alive, and the presidential Blue House said in a statement Saturday it 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". He is a longtime mentor to Kim Jong-un and might now be the most powerful North Korean who is not a member of the Kim dynasty.
"The confrontation is likely to continue as there's a low chance that the United States will change its position", Mr Shin said.
Trump had also dangled the possibility of a third summit.
As the new president of the Supreme People's Assembly, Choe is technically head of state, although real power in North Korea is wielded by Kim and his immediate family.
The remark means that the North can not accept Washington's so-called "big deal" proposal that calls for dismantling all of its nuclear, missile and other weapons programs before any rewards, such as sanctions relief, are given to Pyongyang.
Ms Choe Son Hui, another official who has played a highly visible role in the USA talks, was promoted to the job of first vice foreign minister, a move that could give her even more sway if discussions resume.
"A third Summit would be good in that we fully understand where we each stand", the USA president said in a pair of Saturday morning tweets.