Trump told reporters traveling home to Washington with him aboard Air Force One from Argentina that "three sites" were in consideration for the meeting, a follow-up to their historic June summit in Singapore.
"President Trump asked me to forward to (Kim) these messages; he has a very friendly view of Chairman Kim Jong Un and likes him".
Seoul said Moon and Trump talked denuclearization on the sidelines of the G20 meeting.
"As such, he asked me to tell Chairman Kim that he wants to implement the rest of their agreement together and he will fulfill Chairman Kim's wishes".
A South Korean train is entering North Korea for the first time since 2007.
And days before the meeting between Xi and Trump in Buenos Aires, an adviser to China's foreign ministry said it would be "impossible" for North Korea to completely give up its nuclear weapons, and the US should instead focus on more incremental progress.
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The Gators are 0-4 all-time against the Wolverines with the teams meeting for the first time at the end of the 2002 season. Mullen would also become the first coach in SEC history to garner 10 wins in his first season as head coach.
The US has given its support to the rail survey, but Mr Moon will have to have a number of hard conversations if he's to persuade President Donald Trump that easing sanctions at this stage would be worthwhile so the project can go ahead, our correspondent says. Trump has said he hopes to meet with Kim early next year in a bid to give the talks a boost.
If Kim, a third-generation hereditary ruler, visits Seoul, he would be the first North Korean leader do so since the end of the 1950-53 Korean War.
The defection came just over a year after another North Korean soldier dashed across the border in the truce village of Panmunjom under a hail of gunfire from his former comrades. Moon said a visit would convey Kim's commitment to nuclear disarmament, better inter-Korean relations and world peace.
Trump has frequently described a warm personal relationship with Kim, arguing that this rapport would help him succeed at a diplomatic breakthrough that has eluded USA presidents since the 1950s.
South Korean soldiers escorted the defector to safety after finding him moving south of the eastern side of the military demarcation line that bisects the Koreas, South Korea's Joint Chiefs of Staff said in a statement.
The history of the inter-Korean railways: Before the Korean War, two rail systems connected the northern and southern parts of the Korean Peninsula, per AFP. "We have a good relationship", Mr Trump said, adding that at some point he will invite Mr Kim to the US. "The trains running on the track will also carry peace and prosperity with them to North East Asia and the world", South Korea's Yonhap news agency quoted him as saying.