Moon made the remark onboard his presidential plane that Trump asked him during their meeting Friday at the G20 in Buenos Aires to personally convey the message to Kim, reports Efe news.
"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," he said.
Trump and Kim agreed in June at their historic summit in Singapore that they would work towards the denuclearization of the Korean Peninsula in exchange for Washington ensuring the regime's survival.
Trump confirmed on Sunday his intention to hold a second summit with Kim, which could take place in January or February next year and for which three places were being considered.
However, the stalled talks on denuclearization, motivated by the lack of a specific guideline emerging from the Singapore agreements, has delayed the process to set the second meeting up.
The US has demanded North Korea take concrete action towards denuclearization, while the regime calls for a peace treaty to be signed in order to guarantee security.
Trump highlighted during the meetings he held on the sidelines of the G20 summit in Buenos Aires with Moon and Japanese Prime Minister Shinzo Abe the necessity to maintain sanctions given that Pyongyang was yet to show any progress in denuclearization.
Putin expressed his readiness to hold a summit with Kim at an "early date" in a message sent on the occasion of North Korea's liberation, coinciding with the 73rd anniversary of Japan's surrender in World War II, North Korea's state news agency KCNA reported.
"I affirm that I am ready to meet you at an early date to discuss urgent issues of bilateral relations and important matters of the region," Putin said.
Earlier in May, the Russian President invited Kim to the Eastern Economic Forum in the Russian coastal city of Vladivostok between September 11 and 13, but Pyongyang was yet to respond.
Putin also invited his South Korean counterpart Moon Jae-in, Chinese counterpart Xi Jinping and Japanese Prime Minister Shinzo Abe to the forum.
If Kim and other leaders decided to attend, it would be an unprecedented international gathering in which leaders from five of the six countries that have been working on nuclear disarmament meet.
An invitation was sent by the Indonesian Coordinating Human Development and Cultural Affairs Minister Puan Maharani to the President of the North Korean Supreme People's Assembly, Kim Yong-nam, during a meeting in Pyongyang earlier this week, Efe news reported.
Puan tweeted that "the Asian Games is a time for friendship between countries". Widodo also invited South Korean President Moon Jae-in to the inaugural ceremony.
North and South Korea agreed in June during a meeting at the border to walk under the same flag at the opening ceremony of the Asian Games and compete with joint teams in some categories.
The two Koreas, technically still at war, walked together at the opening of the Winter Olympic Games in February in the South Korean county of PyeongChang which marked the first steps towards the recent rapprochement between the two Koreas and Washington.
"I probably have a very good relationship with Kim Jong-un," Trump said in an interview about the first year of his presidency with The Wall Street Journal newspaper.
"I have relationships with people. I think you people are surprised," said the US President.
Trump, however, declined to confirm whether or not he had spoken with Kim, Efe news reported.
"I don't want to comment on it. I'm not saying I have or haven't. I just don't want to comment," he replied, when asked by WSJ if he had contacts with the North Korean leader.
Trump has been of late more open, on more than one occasion, to having a dialogue with North Korea.
On January 10, the White House cited Trump's willingness to hold talks "at the appropriate time, under the right circumstances" with Pyongyang.
The US has had no official contacts with North Korea for years. Over the past few month Trump had hostile exchanges Kim, who has repeatedly been mocked by Trump as a "rocket man" over his weapons programs.
According to the US President, "all these harsh comments were part of a strategy".
"You'll see that a lot with me," Trump said of his belligerent remarks towards the North Korean leader, but added "then all of a sudden somebody's my best friend.
"I could give you 20 examples. You could give me 30. I'm a very flexible person."
On January 9, South and North Korea held a "historic" meeting after a gap of two years. The delegations agreed to ensure "the safety and success" of the upcoming PyeongChang Winter Olympics in the South.
North Korea has expressed its interests in attending the Winter Olympics, in February which marks an important moment of possible rapprochement between the two Koreas after a year of tensions triggered by Pyongyang's continuous weapons tests and the US President's aggressive remarks towards the regime.
When Asked if Kim intended to "drive a wedge" between Seoul and Washington by opening the talks with South Korea, the US President did not rule it out.
"If I were them, I would try," he said, but added "The difference is I'm President, other people aren't. And I know more about wedges than any human being that's lived."
Trump and Kim had been scheduled to hold a working lunch before holding a joint press conference later, but those plans were abruptly abandoned around midday.
"The two leaders discussed various ways to advance denuclearisation and economic driven concepts. No agreement was reached at this time, but their respective teams look forward to meeting in the future," White House Press Secretary Sarah Sanders was cited as saying by CNN.
She added that the two leaders had "had very good and constructive meetings".
The original White House programme for the day had planned for a "Joint Agreement Signing Ceremony" as well as a working lunch for the two leaders, but expectations were abruptly dashed with the cancellation of both.
Kim was seen leaving the Metropole hotel in central Hanoi, while Trump's motorcade also departed from the venue of the summit for the Marriott hotel, where a press conference was scheduled for later.
Reporters at the Metropole hotel were first told that the lunch was delayed, before a White House spokesperson announced there had been "a programme change".
The summit, the second between the two leaders in less than a year, was held to kickstart the stalled denuclearisation of the Korean Peninsula, as agreed at their first meeting in Singapore last year.
Kim and Trump had met earlier on Thursday at the Metropole hotel, during which the North Korean leader had said that he would not have attended the summit if he did not want to denuclearise.
"If I wasn't willing to do that, I wouldn't be here right now," Kim said. He also told reporters that he was feeling good about what could be achieved.
"It's too early to tell but I wouldn't say that I'm pessimistic. What I feel right now, I do have a feeling that good results will come out," Kim told reporters through a translator.
Trump said he was in "no rush" during the negotiation process, saying: "Speed is not important. What's important is that we do the right deal."