"The message is that President Trump has very favourable views toward Chairman Kim and he likes him", Mr Moon told reporters aboard a flight from Argentina to New Zealand, where he started a three-day state visit on Sunday.
"We're getting along very well. We have a good relationship", Trump said.
North Korea entered into agreements with regional powers in 1994 and in 2005 to dismantle its nuclear programme in return for economic benefits and diplomatic rewards, but those deals broke down after Pyongyang clandestinely continued to pursue building weapons of mass destruction.
Interested in North Korea?
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
On board is a team of South Korean experts investigating ways to help North Korea modernise its rail network.
Trump's spokesperson Sarah Sanders said both the countries has "reaffirmed their commitment to achieve the final, fully verified denuclearization" of the North Korea.
Mr Kim and Mr Trump pledged at their first meeting to work towards denuclearisation, although the two sides have since made little progress agreeing on a timeline or concrete steps.
Despite the warming ties, it remains unclear whether the North's leader Kim Jong Un will make his first-ever visit to the South this year, as Seoul is hoping. Kim agreed to travel to Seoul after hosting Moon in Pyongyang in September for their third summit this year.
Following Moon-Trump meet in G20 summit in Argentina, Moon has developed some respect and support for Seoul summit from the U.S. administration- aiming to entice Kim to hold a good pledge.
Wall Street Cheers Ceasefire in U.S.-China Trade War
China imposed a retaliatory 25 percent tariff on imports of cars from the USA over the summer in response to Trump's own tariffs. However, Chinese attacks on USA companies ratcheted up again when the United States first implemented trade tariffs in July 2018.