North Korea's state media has not immediately responded to South Korea's overtures.
Moon has suggested hostile military activities be halted at the inter-Korean border on July 27, the anniversary of the 1953 armistice agreement that ended the Korean War.
Seoul's proposal for two sets of talks indicates President Moon Jae-in is pushing to improve ties with Pyongyang despite the North's first intercontinental ballistic missile this month. "They can signal to the other side what they want, what problems or issues they have with the other side", said Daniel Pinkston, a North Korea analyst with Troy University in Seoul.
The latest in a series of moves by South Korea's new government to try to seek diplomacy with their neighbors to the north, the defense ministry today announced a formal offer for direct talks between the two Koreas' militaries, to be held on Friday in the truce village of Panmunjon, in the demilitarized zone.More news: Turnbull creates home office portfolio in major security overhaul
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There are also reports of increased activity at the Yongbyon uranium enrichment facility that could indicate plutonium production underway in the previous year to further increase the North's nuclear weapons stockpile.
The North's ICBM (intercontinental ballistic missile) launch has stoked security worries as it showed the country could eventually ideal a reliable nuclear missile capable of reaching anywhere in the United States. Nevertheless, the European Union provided no details on dialogue it was going to conduct with Pyongyang. Mr Kim's statement suggested he will order more missile and nuclear tests until North Korea develops a functioning ICBM that can place the entire U.S. within its striking distance.
However, the South said it couldn't fulfill either demand because it's a liberal democracy.
"There is no change in our position that we will put forth effort to (persuade) North Korea to come out for denuclearization by using all possible measures such as sanctions and talks", said South Korean Unification Minister Cho Myung-kyun.