USA officials insist on the complete, verified and irreversible denuclearisation of the Korean Peninsula before sanctions are lifted.
"We share a commitment to maintaining peace, prosperity and stability in our region", Ardern said.
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.
Multiple ships linked to prohibited North Korean activities were identified after a training workshop with 14 Pacific nations in July, a senior U.S. State Department official told Reuters, without identifying how many ships or the countries involved.
Donald Trump wants North Korean leader Kim Jong Un to know that he likes him and will fulfil his wishes, South Korea's president Moon Jae-in said on Sunday, a day after meeting the USA president at an economic summit in Argentina.
Moon said that he and Trump agreed that the sanctions should not be lifted at this point (although the recent agreement to allow a train-crossing of the demilitarized zone to test the tracks in the North constituted an "exception" to the sanctions, which could be a precedent for future similar "exceptions").
"The crisis is still continuing because of North Korea's development of nuclear, missile and weapons program and materials".
If Kim, a third-generation hereditary ruler, visits Seoul, he would be the first North Korean leader do so since the end of the 1950-53 Korean War.
In a speech to the leaders on the last day of the G20, President Moon shared his administration's ambitious goal of cutting the country's reliance on coal and nuclear power and increasing the proportion of electricity it gets from renewable sources, ... something he's promised to raise to 20 percent by the year 2030.
North and South Korea have begun to remove landmines and destroy military bunkers at parts of their common border as part of efforts to improve long-strained relations.
Mr Trump, who met Mr Kim in Singapore in June, said on Saturday that he is likely to meet the North Korean leader for a second time in January or February, with three sites for their meeting under consideration.
A USA think tank said last month it had identified at least 13 of an estimated 20 active, undeclared missile bases inside North Korea, underscoring the challenge for American negotiators hoping to persuade Kim to give up his weapons programs.
Three sites are being considered, he added.