North Korea is seeking security guarantees from the United States and relief from worldwide sanctions.
They reviewed the recent developments related to the DPRK and discussed how best to work with the worldwide community, including South Korea, to continue to put pressure on the DPRK until it implements its commitment to denuclearize.
South Korea's president said Tuesday no timeframe has been set for a historic visit to Seoul by North Korean leader Kim Jong Un despite both hoping it will take place this year.
Former UN Secretary-General Ban Ki-moon urged North Korean leader Kim Jong Un on Monday to take concrete steps toward complete denuclearisation and gain the trust of the worldwide community if he wants sanctions lifted.
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.
"There is a possibility that Chairman Kim Jong-un's visit to Seoul may be made within this year, but there's more important things than the timing", he said in translated remarks during a visit to New Zealand.
President Moon Jae-in met with New Zealand's main opposition leader on Monday during his state visit to the country and discussed ways to advance bilateral cooperation.
December 3, 2018 (EIRNS)-South Korean President Moon Jae-in spoke to the press after the G20, where he had held a half-hour meeting with President Donald Trump.
Trump said earlier in October that his meeting with Kim would be held after the midterm Congressional elections, and the venue of his second meeting with Kim had been narrowed down to "three or four" locations.
Since the initial agreement for denuclearization, the United States has pressed the North Koreans to provide information on the entirety of its nuclear operations.
Moon has also facilitated a series of high-level talks between the USA and North Korea, including a summit between Trump and Kim in June.
"New Zealand's stance on nuclear non-proliferation is clear and it is our hope that 2019 will see further progress towards achieving denuclearisation of North Korea in a complete, verifiable and irreversible manner and an enduring peace on the Korean peninsula", Ardern said in an emailed statement.