Addressing journalists at Camp David, Trump described talks between the two Korean nations as a "big start", insisting that Kim knew Trump was "not messing around".
South Korean opposition parties have struck a cautious tone over the latest developments, warning against making concessions to the North to secure its Olympic participation.
She said: "It's very important that we don't ever let him get so arrogant that he doesn't realize the reality of what would happen if he started a nuclear war".
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He told reporters Friday that Japan will "take all steps necessary", including cooperating with China and Russian Federation, to step up pressure on North Korea so it will give up its nuclear weapons and missile development policies.
A North Korea expert, Yang Mu-jin, a professor at the University of North Korean Studies in Seoul, said Friday that the breakthrough could give both sides the chance to discuss security issues, "make claims and listen to the other".
"Look, right now they're talking Olympics".
"If I weren't involved, they wouldn't be talking about the Olympics right now, they'd be doing no talking", he said.
Moon, a liberal, has been pushing to improve strained ties and restore stalled co-operation projects with North Korea since his inauguration in May, though he joined US -led worldwide efforts to apply more pressure and sanctions on the North.
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The North's message comes a day after the South offered to send a five-member delegation led by Unification Minister Cho Myoung Gyon to the talks scheduled for tomorrow.
The ramped-up momentum for inter-Korean dialogue follows a year of missile and nuclear tests by North Korea as well as an exchange of bellicose comments from Trump and Kim, which raised alarm across the world.
Seoul's Unification Ministry said it had received a fax from Ri Son-Gwon, head of the North's Committee for Peaceful Reunification of Korea, a state agency handling inter-Korean affairs, that said: "We will come to the Peace House at Panmunjom on January 9".
They may not mix, but that doesn't stop them from being frequently tossed into a blender anyway.
The last high-level talks between the two countries took place in a deputy ministerial meeting in December 2015, at the jointly-run Kaesong industrial zone in North Korea.
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Relations between North Korea and the global community have been tense in recent months following a slew of intercontinental ballistic missile launches.