SEOUL, South Korea — President Moon Jae-in of South Korea said on Monday that he wanted to meet again with North Korea’s leader, Kim Jong-un, making the overture three days after Mr. Kim dismissed Mr. Moon’s mediating efforts between the North and the United States as “officious.”

“Now is the time to begin the preparations in earnest for an inter-Korean summit,” Mr. Moon said. “As soon as the North becomes ready, I hope the two Koreas will be able to sit down together, regardless of venue and form, and hold detailed and substantive talks on how to achieve further progress that goes beyond the previous two summits between Chairman Kim and President Trump.”

Mr. Moon met with Mr. Kim three times last year to help pave the way for Mr. Kim’s first summit meeting with Mr. Trump in Singapore last June and their second meeting in Hanoi, Vietnam, in late February. Mr. Moon has repeatedly argued that a nuclear disarmament deal is possible, insisting that Mr. Kim is willing to give up his weapons and focus on economic growth should Washington provide the right incentives.

But Mr. Moon’s mediating role was cast into doubt when the Hanoi summit ended abruptly without a deal, and he has since been struggling to assert his relevance. Despite Mr. Moon’s optimism, both Mr. Trump and Mr. Kim found in Hanoi that their terms on denuclearizing North Korea remained too far apart for a compromise.