Earlier Tuesday, Moon spoke of Kim's positive response to earlier South Korean dialogue overtures and ordered officials to study how to restore talks with North Korea and get the North to participate in the Olympics.
"We look forward to candidly discussing interests from both sides face-to-face with North Korea along with the North's participation in the Pyeongchang Winter Olympics", Unification Minister Cho Myong-gyon told reporters.
The New Year's address is an annual event in the North and is watched closely for indications of the direction and priorities Kim may adopt in the year ahead.
South Korea's presidential office welcomed Kim's olive branch, saying that Seoul is open to talks with the North "at any time, in any place and in any format".
But the North has not answered a call by the South since it severed the channel in February 2016 in an angry protest against Seoul's shutdown of an inter-Korean industrial complex.
North Korean leader Kim Jong Un makes a speech at 5th Conference of Cell Chairpersons of the Workers' Party of Korea (WPK) on December 23 in this photo released by North Korea's Korean Central News Agency (KCNA) in Pyongyang December 24, 2017.
Beyond his talk of a "nuclear button", Kim Jong-un's most diplomatically significant remark in the address concerned inter-Korean relations.
Mullen's warning comes as tensions continue to increase between North Korea and the global community.
The president vowed to destroy North Korea if the country continued to threaten the US, speaking during his maiden speech at the United Nations earlier this year.
Despite those declarations, Kim then called for accelerated production of nuclear warheads and ballistic missiles.
But Mullen also said that Trump has disrupted long-held views of the US role in the world, and that America's enemies benefit from that uncertainty.
After reiterating that his country had "completed" its nuclear deterrent against the United States - a fact that its state media proclaimed following the successful flight-test of the Hwasong-15 intercontinental-range ballistic missile on November 29 - Kim added that he had a "nuclear button" physically installed on his desk in his office.
South Korean Unification Minster Cho Myoung-gyon says the South proposes the two Koreas meet January 9 at the border village of Panmunjom to discuss Olympic cooperation and how to improve overall ties.