Beijing has been joined by Moscow in its "suspension-for-suspension" proposal, in which North Korea suspends weapons tests, while South Korea and the U.S. defer further large-scale military exercises.
North Korea had accused the South of kidnapping the waitresses, while Seoul said they fled the hermit kingdom on their own.
Mr Moon suggested earlier this month hostile military activities at the border be ended on 27 July, the anniversary of the 1953 armistice agreement that ended the Korean War.
On the separate offer to hold family reunions, the Korean Red Cross asked the North to hold relevant talks on August 1 at the Peace House, a South Korea-controlled building in the truce village, known as the equivalent to North Korea's Tongilgak. Last time Seoul and Pyongyang held intergovernmental talks in late 2015.
After the North announced the successful launch of an intercontinental ballistic missile on July 4, Harris said it is getting "closer to being able to deliver a nuclear-equipped missile" to the USA mainland.
South Korea proposed military talks and family reunions with North Korea on Monday to reduce the increasing potential for conflict over Pyongyang's continued development of nuclear weapons and ballistic missiles. "[The government] is looking forward to a positive response from the North Korean side".
The request to hold talks is the first such formal overture by the new South Korean administration.
It was unclear how North Korea will react since it remains suspicious of new South Korean President Moon Jae-in's outreach to it.
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".
A separated family reunion applicant corrects his information at the headquarters of South Korea's Red Cross in Seoul on Monday.
This offer of a dialogue is the first since the arrival to power in may of president Moon Jae-In, perceived as more open to negotiation than his predecessor.
Family reunification and humanitarian improvement are other goals alongside the de-escalating the nuclear and military conflict between the two Koreas. Seoul however says the group-12 waitresses and a restaurant manager- had defected to the South a year ago.
North Korea, which has conducted five nuclear tests since 2006, including two since the beginning of 2016, was successful on July 4, his first shot of missile intercontinental.
The proposals were separately announced to local media by the Ministry of National Defense and the Korean Red Cross, the latter of which organizes the family reunions.