While the North has repeatedly stressed these reciprocal measures should involve relief from global sanctions, President Moon on Tuesday said he believed there were other ways for Washington to build trust.
Kim sharply raised tensions with nuclear and missile tests last year, but suddenly reached out to South Korea and the United States this year with a vague nuclear disarmament pledge.
A visit by North Korean leader Kim Jong Un to Seoul before the end of the year remains possible, South Korean President Moon Jae-in reiterated to press on Tuesday.
Former UN Secretary-General Ban Ki-moon has urged North Korean leader Kim Jong-un to take concrete steps toward complete denuclearization 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.
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. The pair "reaffirmed their commitment to achieving the final, fully verified denuclearisation" of North Korea, Trump's spokeswoman Sarah Sanders said. "U.S. President Donald Trump also holds a very positive view of the current situation", he confirmed.
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President Ramaphosa received a resounding applause after being introduced by Oprah Winfrey . It said it urged South African police "to act swiftly and efficiently".
President Trump is "very friendly" towards Kim Jong-un and wants to make his wishes come true in return for denuclearisation, President Moon of South Korea said on his return from the G20 summit.
After their third meeting in North Korea's capital, Pyongyang, in September, Moon said Kim agreed to make a reciprocal visit to Seoul in 2018.
After Trump and Kim met in Singapore in June, negotiations between Washington and Pyongyang have stalled due to differences on the scale of denuclearization, USA sanctions, and whether to issue a war-ending declaration, among others.
Moon said a second summit between Kim and Trump will prove to be the "most critical moment" for North Korea's denuclearisation. The two countries signed a free-trade deal in 2015.
Moon's programme in Auckland includes a ceremony of welcome and a state lunch hosted by Governor-General Dame Patsy Reddy at Government House, a wreath laying ceremony and tour of Auckland War Memorial Museum, and an engagement with the Korean community in Auckland.