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.
Previously, North Korean state media announced that Kim Jong-un said he expected significant results at his second meeting with Donald Trump.
In June, Trump and Kim opened up dialogue on denuclearization of the Korean peninsula after months of trading military threats and pointed barbs.
"I'd like to really advise the North Korean leader that the world is ready" to help his country, Ban told The Associated Press, specifically citing South Korea, the U.S., China, Japan and Russian Federation.
When asked about a future meeting, Trump said: "I think we´re going to do one fairly (soon) - into January, February, I think".
Sanders said both leaders agreed on the necessity for "maintaining vigorous enforcement of existing sanctions to ensure the DPRK (Democratic People's Republic of Korea) understands that denuclearization is the only path".
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The postponement stirred concerns that nuclear talks between the US and North Korea had again bogged down, as Kim's regime seeks sanctions relief before taking key disarmament steps.
The primary goal of discussions between the US and North Korea remains the creation of "a nuclear-free Korean Peninsula", said a White House statement issued Saturday following Trump's meeting in Argentina with Chinese Present Xi Jinping. The South Koreans have expressed hope the visit - which would be a first for a North Korean leader - could happen by the end of this year.
But differences have emerged between Washington and Seoul on how to proceed with Kim, as the dovish Moon has long favored engagement with the North.
"We're getting along very well, we have a good relationship", Trump also said.
However, after hosting their third summit meeting with Moon in Pyongyang in September, Kim has agreed to visit Seoul.
Last November, a North Korean soldier was critically wounded in a jointly controlled area after he fled to the South amid a hail of bullets fired by his former comrades.