The current railway project has also faced delays over concerns it could violate United Nations sanctions imposed on the North over its nuclear and missile programmes.
The White House said in a statement on Saturday after Trump's meeting with Chinese President Xi Jinping that they and Kim will strive "to see a nuclear free Korean Peninsula".
"Through the one connected railway, the South and the North will prosper together and the ground for peace on the Korean Peninsula will be consolidated".
In November previous year, a North Korean soldier drove to the heavily guarded border at speed and ran across under fire from his own comrades.
The Unification Ministry said the North will attach its own rail cars to the vehicles, but it was unclear how many.
Search Underway After Two US Marine Aircraft Crash Off Japan
The crash comes just a month after a US F/A-18 Hornet from the aircraft carrier USS Ronald Reagan crashed into the sea southwest of Okinawa.
The six-carriage train is transporting 28 South Koreans including railway engineers and other personnel, and carrying 55 tonnes of fuel and an electricity generator. Add North Korea as an interest to stay up to date on the latest North Korea news, video, and analysis from ABC News.
The Koreas and the U.S. -led U.N. Command recently finished removing firearms and troops from the jointly controlled area at the border village of Panmunjom, and eventually plan to allow tourists to freely move around there.
President Trump said Saturday that his next face-to-face meeting with North Korean leader Kim Jong Un will likely be held in January or February, at a site yet to be determined. The former South Korean foreign minister said on Monday "this is the time" to achieve Pyongyang's denuclearization after almost a quarter century of unsuccessful efforts, adding that the North and the world should not lose this opportunity. "I'd like to really advise the North Korean leader that the world is ready" to help his country, Ban told AP, specifically citing South Korea, the US, China, Japan and Russian Federation.
US Secretary of State Mike Pompeo was due to meet with a top North Korean official in early November, but the meeting was abruptly put off, with North Korea insisting that Washington ease sanctions. Kim agreed to travel to Seoul after hosting Moon in Pyongyang in September for their third summit this year. They also announced measures to reduce conventional military threats, such as creating buffer zones along their land and sea boundaries and a no-fly zone above the border, removing guard posts and demining sections in border areas.