North Korea has sought security guarantees from the United States and relief from worldwide sanctions.
Moon, who is hoping to host Kim soon on the first ever trip to Seoul by a North Korean leader as agreed earlier this year, said Trump had asked him to pass on a message.
Former UN Secretary-General Ban Ki-moon urged North Korean leader Kim Jong Un on Monday to take concrete steps toward complete denuclearisation and gain the trust of the worldwide community if he wants sanctions lifted.
The White House said in a statement on Saturday after Mr Trump's meeting with Chinese President Xi Jinping that they and Mr Kim will strive "to see a nuclear-free Korean peninsula".
Moon spoke shortly after a meeting with New Zealand Prime Minister Jacinda Ardern where they discussed how to secure a denuclearised Korean peninsula.
After their third meeting in North Korea's capital of Pyongyang in September, Moon said Kim agreed to make a reciprocal visit to Seoul this year.
The International Space Station Welcomes Its New Residents NASA spokesperson Gary Johnson described the mission as "textbook launch and insertion into orbit" during the liftoff commentary. The three astronauts will begin an expected six and a half month stay aboard the ISS, the Russian Roscosmos space agency said.
Trump has frequently described a warm personal relationship with Kim, arguing that this rapport would help him succeed at a diplomatic breakthrough that has eluded United States presidents since the 1950s.
Ban said he hopes their second summit will occur at a time when the worldwide community feels confident about North Korea's commitment to denuclearisation.
Moon has met Kim three times this year.
"Until such time as North Korea abides by its global obligations, full implementation of the United Nations Security Council sanctions resolutions will be essential".
Moon said a visit by Kim to the South would also improve US-North Korea relations as Kim and President Donald Trump look to follow up their June summit in Singapore. Their two previous summits were held at the countries' shared border village of Panmunjom.
"This will be the first time that the North Korean leader will visit Seoul since Korea was divided into two countries", he said. But both Moon and Trump say fears of war on the Korean Peninsula have eased sharply.