South Korean President Lee Jae Myung vowed on Tuesday to end a "vicious cycle of unnecessary military tensions" with North Korea, with the aim of achieving peaceful coexistence and shared growth.

In his first speech to the United Nations General Assembly, Lee emphasised his vision of a "phased solution" to the North Korean nuclear issue, "based on a cool-headed perception that denuclearisation cannot be achieved in the short term."

On Sunday, the leader of nuclear-armed North Korea, Kim Jong Un, rejected any phased plan, saying recent overtures from Washington and Seoul for dialogue were disingenuous because their fundamental intent to weaken Pyongyang remained unchanged.

Kim said there was no reason to avoid talks with the US if Washington stopped insisting he give up his nuclear weapons, but he would never abandon the nuclear arsenal to end US-led sanctions.

Lee said Seoul would "consistently seek a path to reduce military tensions and restore inter-Korean trust" and pointed to recent actions like halting propaganda leaflets and loudspeaker broadcasts to North Korea.

"By gradually expanding inter-Korean exchanges in cooperation, we will pave the way for sustainable peace," he said.

On Sunday, the North Korean leader said he would never engage in dialogue with Seoul.

However, Kim said he had "fond memories" of US President Donald Trump, with whom he held several summits in the US leader's first term, engagement that collapsed over US denuclearisation demands.

Trump said last month he wanted to meet Kim this year, but in a wide-ranging 55-minute speech to the UN on Tuesday, the US leader made no mention of North Korea. (Reuters)