Relations between North and South remain tense decades after the war
Mr Lee said the project would be a good way to mark the 60th anniversary of the start of the war later this year. The North has yet to respond to the idea.
North Korea has previously allowed US military teams to search for remains of soldiers killed during the war.
But there has not yet been any joint project for South Korean remains.
The two Koreas technically remain at war, because the 1950-1953 conflict ended in a truce, not a peace treaty.
Turning point?
"South Korea will not forget those who gave their lives on foreign soil," Mr Lee said.
The Korean War ended in a truce in 1953
The remains of about 39,000 soldiers from the South, killed during the war, are thought to be in North Korea, according to the Defence Ministry in Seoul.
Mr Lee made his proposal during a speech in which he called for a new chapter in relations between the two neighbours.
He said he was seeking "a turning point", and again urged the North to rejoin six-party negotiations aimed at ending its nuclear programmes.
"For this, there needs to be a standing dialogue organisation between the South and the North," he said, without elaborating.
Mr Lee's speech follows a call by North Korea on Friday for an end to hostile relations with South Korea's key ally, the United States, and a nuclear-free peninsula.
In response, a US State Department official said North Korea should show its good faith by returning to six-party talks.
Pyongyang pulled out of the talks last April following widespread condemnation of a long-range missile launch. It subsequently conducted a nuclear test, which drew UN sanctions.
But in December, North Korea said it would work with the US to "narrow remaining differences" following a visit to Pyongyang by US President Barack Obama's special envoy Stephen Bosworth.
JPAC, the US unit charged with finding remains of war dead, sent 33 missions to North Korea from 1996 to 2005, leading to the identification of more than 20 sets of remains.
US recovery missions ended in 2005 amid rising tensions over nuclear negotiations with North Korea; there are about 8,100 US servicemen not accounted for from the Korean War, according to JPAC.
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