Kim left the North Korean capital of Pyongyang at around 5 p.m. (0800 GMT) in an armored train, Russia's TASS news agency said, citing a North Korean diplomatic source.
Kim Jong Un is slated to meet with President Trump for the second time in Hanoi on Wednesday, and again on Thursday.
Some media speculated about Kim Jong Un traveling to Beijing for a meeting with his ally, Chinese President Xi Jinping, before continuing towards Vietnam.
If the train does trundle all the way to Hanoi carrying the North Korean leader, it will mean a almost 4,000-km, 60-hour journey on board for Kim.
Trump met Kim last June in Singapore, the first meeting between leaders of the two countries since the end of the Korean War in 1953.
The North's official Korean Central News Agency reported early Sunday that Kim was accompanied by Kim Yong Chol, who has been a key negotiator in talks with the USA, and Kim Yo Jong, the leader's sister.
Kim's train will reportedly stop at the Vietnamese border station of Dong Dang, after which he will proceed by vehicle to Hanoi, with local authorities planning to close the stretch of road that he will travel on between the station and the capital.
Kim will also make an official goodwill visit to Vietnam "soon", KCNA added.
In their Singapore summit, Mr. Kim Jong-un and Mr. Trump pledged to work towards peace between their countries and for the denuclearisation of the Korean peninsula.
Kim wants to leverage his nuclear and missile program for economic and security benefits, but doubts remains as to whether he is prepared to deal away an arsenal that he may see as his strongest guarantee of survival.
Last year, North Korea suspended its nuclear and long-range missile tests and unilaterally dismantled its nuclear testing ground and parts of a rocket launch facility without the presence of outside experts, but none of those steps were seen as meaningful cutbacks to the North's weapons capability.
Hanoi has been gearing up for the summit with beefed up security. Workers decorated the area with flowers and flags of North Korea, the United States, and Vietnam.
The preferred location for the summit is the Government Guesthouse, a colonial-era building in central Hanoi, three sources said Wednesday.
At the same time, some foreign media organizations received a note from Vietnam's foreign ministry prohibiting live broadcasts in and around the Melia hotel and the scenic Hoan Kiem lake in the centre of Hanoi.
However, on late Friday, a Kim lookalike, the Hong Kong-based impersonator who uses the name Howard X, posted on Facebook that about 15 police or immigration officers demanded a mandatory "interview" with them following a talk they gave at the state-run VTC station.