Trump said on Air Force One while returning to Washington from Argentina that his next meeting with Kim would likely happen in January or February. He said, "We are getting along very well. We have a good relationship".
Mr Trump added that at some point he would invite Mr Kim to the United States.
The Trump-Kim summit in June of this year was created to help ease the tensions between the two nations - though critics have questioned whether North Korea has complied with agreements reached at the historic meeting.
To improve long-strained relation between the South and North Korea, they both have started their mission by removing landmines and also destroying military bunkers at their border.
"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.
A South Korean train runs on the rail track which the two Koreas hope to eventually use as an international transport link near to the border villag
U.S. President Donald Trump and North Korea's leader Kim Jong Un walk together before their working lunch during their summit at the Capella Hotel on the resort island of Sentosa, Singapore June 12, 2018.
On Friday, Trump discussed the situation with South Korean President Moon Jae-in on the sidelines of the G20 summit. "We talked about the need for free, fair and reciprocal trade, and our commitment to the final, fully-verified denuclearization of #DPRK". Re-connection of train line are in progress with another rail which links across the border.
At the U.N. General Assembly in NY that year, for example, Trump said the USA would "totally destroy" North Korea if it continued with its provocative actions.
However, after hosting their third summit meeting with Moon in Pyongyang in September, Kim has agreed to visit Seoul.
Moon said he and Trump agreed during their meeting that Kim's trip to Seoul would play a "very positive role" in U.S.
Former U.N. Secretary-General Ban Ki-moon urged North Korean leader Kim Jong Un on Monday to take concrete steps toward complete denuclearization and gain the trust of the global community if he wants sanctions lifted.