A South Korean train rolled across the heavily militarized frontier into North Korea for the first time in a decade Friday, as Seoul pushed ahead with a plan to reunite the two railway networks despite heavy United Nations sanctions.
"I think we're going to do one fairly (soon) - into January, February, I think", Trump said in remarks carried by the AFP news agency.
The White House said in a statement after Trump's meeting with Chinese President Xi Jinping that they and Kim will strive "to see a nuclear-free Korean Peninsula".
When asked Saturday if he would ever host Kim in the United States, the Republican president replied: "At some point, yeah".
The two leaders signed a vaguely worded document on denuclearisation of the peninsula, but progress since has stalled as Washington and Pyongyang spar over the meaning of the document.
Former US president George HW Bush dead at 94 - son
Bush oversaw the opening of his presidential library in College Station, Texas, and was widely honored as an elder statesman. He was the 41st USA president between 1989 and 1993, after serving two terms as vice-president to Ronald Reagan.
A South Korean train carrying 28 government officials and railway experts departed for North Korea on Friday morning to start a joint survey with Pyongyang on North Korean tracks, a major step in a cross-border infrastructure project aimed at modernizing North Korean train lines and connecting them to the South to aid Pyongyang's economic development.
Trump has remained optimistic but has insisted economic sanctions and other US -led efforts to pressure the communist state into giving up its nuclear weapons will continue until concrete progress is made. The administration of South Korean President Moon Jae-in appears determined to host Kim in Seoul before the end of the year. It was essential that worldwide sanctions pressure be maintained on North Korea until its complete denuclearisation was achieved, he said.
In recent months, however, the two sides have begun to remove landmines and destroy military bunkers at parts of the border as part of efforts to improve long-strained relations.
Since the Korean leaders held talks for the first time in April, they have been trying to ease tensions along the border. But prospects of a fourth Moon-Kim meeting have recently dimmed, with negotiations on denuclearising the North grinding to a halt.
The two leaders made a vague commitment on "denuclearization" of the Korean Peninsula during the summit, but glossed over how to make it happen.
But while inter-Korean relation-building has continued apace over the past year, talks with the U.S. have stalled.