The Red Cross said it hoped for "a positive response" from its counterpart in the North, hoping to hold family reunions in early October.
If realised, they would be the first in two years.
Millions of families were separated by the conflict that sealed the division of the two countries.
Many died without getting a chance to see or hear from their families on the other side of the heavily fortified border, across which all civilian communication is banned.
With the passage of time, the number of survivors has diminished, with only around 60,000 left in the South.
Moon, who took power in May, has advocated dialogue with the nuclear-armed North as a means of bringing it to the negotiating table as tensions soar over its weapons ambitions.
But Pyongyang has staged a series of missile launches in violation of UN resolutions, most recently on July 4 when it test-fired its first ICBM, a move which triggered global alarm and a push by President Donald Trump to impose harsher UN sanctions on the country.