"We request military talks with the North on July 21 at Tongilgak to stop all hostile activities that raise military tension at the military demarcation line", Vice Defense Minister Suh Choo-suk told a media briefing Monday.
North Korea didn't immediately respond to the latest offers.
After the ICBM launch, Kim said he would never negotiate over his weapons programs as long as USA hostility and nuclear threats persist.
The offer comes after the North claimed to have conducted the first test of an intercontinental ballistic missile (ICBM) earlier this month, and said it had mastered the technology to mount a nuclear warhead on the missile.
Outside experts believe the South Korean broadcasts and leaflets sting in Pyongyang more because the authoritarian country worries that the broadcasts will demoralize front-line troops and residents and eventually weaken the grip of absolute leader Kim Jong Un.
If the North accepts, it will be the first time since 2015 to hold such talks.
Moon's government proposed two sets of talks to discuss how to dial down tensions and resume reunions of aging Koreans separated by the 1950-53 Korean War.More news: Elon Musk: Governments better get ready for the robots
The South Korean Red Cross suggested talks be held on August 1, with possible reunions over the Korean thanksgiving Chuseok holiday, which falls in October this year. The last reunions were held in October 2015 at a resort at Mount Kumgang on North Korea's east coast.
This computer generated image shows South Korea and North Korea shaking hands.
"The way to a peaceful Korea will not be smooth with the North's missile development and ever-changing worldwide affairs", Father Lee said.
But experts said the resumption of the event may take a bumpy road since the communist state continued in the commentary to demand South Korea repatriate 12 female North Koreans who worked at a Pyongyang-run restaurant in China and defected to Seoul en masse a year ago.
"The fact that we wish to take on a leading role in resolving this (North Korean) issue has already been understood at the summit with the United States and the Group of 20 summit meetings", Cho said on Monday.
The North and the South have been divided since Korea's liberation from the Japanese at the end of World War II.
China's foreign ministry welcomed Seoul's peace gesture, saying it hopes "the two sides will move in a positive direction to. break up the stalemate and to relaunch the dialogue and negotiations".