But it's unclear if those talks will be realized. Ties between the Koreas have plunged over the North's expanding missile and nuclear programs and the hard-line policies of Moon's conservative predecessors. Official talks between the sides have not been held since December 2015.
David Petraeus, the retired general and former Central Intelligence Agency director, said the US needs to continue tightening sanctions on North Korea to get the country's "attention" and persuade it to accept a diplomatic accord over its nuclear arsenal. After the North fired an ICBM on July 4, Moon quickly condemned the launch as a "reckless" and "irresponsible" provocation.
The Red Cross said it hoped for "a positive response" from its counterpart in the North, hoping to hold family reunions in early October.
North Korea is believed to possess hundreds of missiles capable of striking South Korea and Japan.
Gasoline and diesel prices have reportedly jumped in North Korea, after its largest crude supplier China halted fuel sales.
During a period of detente from 1998 to 2008, there was a flurry of talks and exchange programs between the two Koreas.
Speaking in Brussels, Mr Johnson said: "We remain absolutely determined to try to get the North Koreans to see sense and to continue to put pressure on the regime in Pyongyang and we think the best way to do that is, of course, is to put pressure on the Chinese and we are seeing some progress there but more to be done".
The North severed the West Sea hotline in protest against the Park Geun-hye government's shutdown of the Kaesong factory park in February 2016.
The proposed meetings, if realised, would be a "rare opportunity to ease tension that has built up for 10 years", said Cheong Seong-chang, analyst at the Sejong Institute, a thinktank. "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.
Tongilgak is a North Korean building at the Panmunjom truce village on the border used in the past for inter-Korea talks.
Numerous estimated 60,000 South Koreans that have signed up to participate in these sporadically held inter-Korean reunions are elderly and have had virtually no contact with relatives living in the repressive North where contact with the outside world is highly restricted.
South Korea did not disclose what specifically it wanted to discuss if military talks were held. We do have some bargaining chips with Beijing in this regard including a warning that we may otherwise extend diplomatic recognition to Taiwan and provide them with significantly more military aid.