"And the message was that President Trump has a very friendly view of Chairman Kim and that he likes him, and so he wishes Chairman Kim would implement the rest of their agreement and that he would make what Chairman Kim wants come true", Mr Moon told Yonhap News Agency on Sunday, while en route to New Zealand.
Ban, a former South Korean foreign minister, said "this is the time" to achieve North Korea's denuclearisation after almost a quarter century of unsuccessful efforts, and that the North and the world should not lose this opportunity.
US Secretary of State Mike Pompeo was due to meet with a top North Korean official in early November, but the meeting was abruptly put off, with North Korea insisting that Washington ease sanctions. Moon spoke to reporters on Saturday aboard his presidential plane en route to New Zealand from Argentina, where he met Trump on the sidelines of a Group of 20 Nations summit. "We have a good relationship", Trump told reporters aboard Air Force One on his return from a G20 summit in Argentina.
"We're getting along very well".
Moon said he and Trump agreed that a visit by Kim to the South Korean capital would give a boost to plans for a second U.S. But progress has been slow, with the two countries sparring over the exact meaning of the agreement.
If that happens, he would be the first North Korean leader to do so since the end of the 1950-1953 Korean War, though the plan is still unclear.
North and South Korea have begun to remove landmines and destroy military bunkers at parts of their common border as part of efforts to improve long-strained relations.
Trump and Moon announced after their landmark summit in Singapore in June that they had agreed to work towards the denuclearisation of the Korean peninsula, although the two sides have since made little progress agreeing on a timeline or other concrete steps.
A US think tank said last month it had identified at least 13 of an estimated 20 active, undeclared missile bases inside North Korea, underscoring the challenge for American negotiators hoping to persuade Kim to give up his weapons programs.
He did not specify which sites were under consideration, saying they had not been determined yet.