It's also seen as a litmus test of whether the Kim Jong-un regime is interested in changing the course of belligerence.
The election in May of Mr. Moon, a liberal human-rights lawyer, ended decades of conservative dominance in South Korea.
Meanwhile, Moon has said peace would require the complete, verifiable and irreversible denuclearization of the Korean Peninsula. "The President has directed us to not do that and to prepare a range of options, including a military option, which nobody wants to take". After the North fired an ICBM on July 4, Moon quickly condemned the launch as a "reckless" and "irresponsible" provocation.
The U.N. Security Council is working on a new resolution to ratchet up pressure on Pyongyang for the latest missile firing.
The Defense Ministry offered talks with Pyongyang counterparts on Monday, while the National Red Cross floated reunion events for families split by decades of hostilities.
"In order to suspend hostile activities at the military demarcation line, we propose holding an inter-Korean military talk at Tongilgak", the North Korean building inside the truce village of Panmunjeom, the ministry said in a statement.
If the communist state agrees, it would be the first inter-Korean military contact since October 2014.
Moon, who took power in May, has advocated dialogue with the nuclear-armed North to bring it to the negotiating table and vowed to play a more active role in global efforts to tame the South's unpredictable neighbour.
North Korea had accused the South of kidnapping the waitresses, while Seoul said they fled the hermit kingdom on their own.
North Korea, which usually communicates via state-run media, has not publicly responded to either proposal.
Seoul expects to achieve another goal - restoring direct communication channels with the North, all of which have been severed since the shutdown of a joint industrial complex in Kaesong early previous year.
The unification, military and other relevant ministries are discussing the plan one day after Pyongyang first reacted to Moon's suggestion, according to the official who asked not to be named de to sensitivity of the issue. Seoul is expecting a positive response from Pyongyang, which has proposed a halt to hostile activities at the MDL numerous times in the past.
Washington and Seoul have further angered Pyongyang by the controversial deployment of the so-called Terminal High Altitude Area Defense (THAAD) missile system to the restive peninsula to counter what they call the North's "threats".