"We are now considering all possibilities in discussing reciprocal visit by the chairman", Cheong Wa Dae spokesperson Kim Eui-kyeom said Monday, taking a step back from arranging Kim's trip within the year.
"President Trump asked me to forward to (Kim) these messages; he has a very friendly view of Chairman Kim Jong Un and likes him".
Mr Moon's comments came one day after he met the US President at the G20 economic summit in Argentina.
At the most recent summit between Moon and Kim in September, the two leaders committed to reviving economic cooperation when possible, voicing optimism that global sanctions could end and allow such activity.
"We're getting along very well".
After his summit with U.S. President Donald Trump on the sidelines of the G20 meeting in Argentina, South Korean President Moon Jae-in took a cautious stance over an inter-Korean project to rebuild a severed cross-border transport artery, saying it may conflict with global sanctions.
When asked Trump about his idea to host North Korean President in the US, Trump said, "At some point, yeah".
"We have a good relationship", Trump told reporters aboard Air Force One on his return flight from the G20 meeting, according to Reuters. A timetable was not given due to a multiple layer of sanctions imposed against North Korea.
The statement said Xi and Trump "agreed that great progress has been made with respect to North Korea".
After whistling twice, a South Korean train engine pulling six rail cars slowly departed toward North Korea's Panmun Station, near the town of Kaesong, where the rail cars will be reconnected to a North Korean engine.
They agreed on the need for "maintaining vigorous enforcement of existing sanctions to ensure the DPRK understands that denuclearization is the only path", Sanders said, using the North's official name, the Democratic People's Republic of Korea.
Prospects for a second summit to overcome a current logjam in negotiations will likely depend on whether either the US or DPRK can show flexibility with regards to offering gestures surrounding sanctions relief or steps towards denuclearization.