Do will make the proposal when he accompanies President Moon Jae-in to the North Korean capital for a third summit with Kim Jong-un next Tuesday, according to South Korea's Yonhap news agency.
"I plan to make this proposal to the North for the settlement of peace" on the Korean peninsula, he was quoted as telling journalists in Tokyo.
"Seoul and Pyongyang would cohost the games", he said.
Joint team at Winter Olympics
The two Koreas discussed the cohosting of the 1988 Seoul Summer Olympics but negotiations broke down over differences on how to share the events.
Pyongyang’s preparations at the time consisted of the construction of an enormous stadium and an athletes' village.
More recently, North and South Korea formed their first-ever unified Olympic team -- a joint women's ice hockey squad -- for February's winter games in the South Korean resort of Pyeongchang, leading to a diplomatic thaw and eventually an unprecedented meeting between North Korean strongman Kim Jong-un and US President Donald Trump.
2020 Tokyo Games, 2030 World Cup
Olympics chief Thomas Bach said this month he was open to talks with the two Koreas about marching and competing together at the 2020 Tokyo Games.
Bach was speaking at the Asian Games in Indonesia, where the two Koreas paraded together at the opening ceremony and formed unified teams for the second time this year.
The two Koreas formed joint teams in rowing, canoeing and women's basketball at the regional Olympics, winning one gold, one silver and two bronze medals.
For the 2030 Fifa World Cup, Do said South Korea also plans to propose a cohosted tournament involving China, Japan and the two Koreas.
"In this way, we could maintain the current momentum for peace and expand peace on the Korean peninsula to the whole region," he said.