Trump is due to arrive in the Vietnamese capital, Hanoi, on Tuesday evening.
Trump will meet Kim for a brief one-on-one conversation on Wednesday evening, followed by a social dinner, at which they will each be accompanied by two guests and interpreters, White House spokeswoman Sarah Sanders told reporters on Air Force One.
She said that would be followed by more meetings between the two leaders on Thursday.
While there is no real expectation that the second meeting will bring a final deal on ridding North Korea of nuclear weapons that threaten the United States, there are some hopes it could lead to a declaration that the 1950-53 Korean War is at last formally over.
But the United States would expect significant movement by Kim towards denuclearisation in return.
In Singapore, Kim pledged to work toward complete denuclearisation of the Korean peninsula, but the vague agreement struck there has produced few concrete results. US Democratic senators and security officials have warned Trump against cutting a deal that would do little to curb North Korea's nuclear ambitions.
Kim travelling from the North Korean capital by train.'s train arrived at the station in the Vietnamese town of Dong Dang after crossing over the border from China, a Reuters witnesses at the station said.
Vietnamese officials were on hand to receive him at the station with a red-carpet including a guard of honour and North Korean and Vietnamese flags flying.
Kim left his train in Dong Dang and set off on the 170 km (105 miles) to the capital, Hanoi, by car.
Both Kim and Trump are also due to hold separate talks with Vietnamese leaders.
US Secretary of State Mike Pompeo also arrived in Hanoi, on Tuesday, a reporter travelling with him said.
Pompeo has been Trump's top envoy in his efforts to improve ties with the reclusive North and has made several trips to Pyongyang to negotiate steps towards ending its nuclear programme.
Trump told reporters he and Kim would have "a very tremendous summit".
Tweeting on Monday, he stressed the benefits to North Korea if it gave up its nuclear weapons. "With complete Denuclearization, North Korea will rapidly become an Economic Powerhouse. Without it, just more of the same. Chairman Kim will make a wise decision!" Trump said.
In a speech on Sunday night, Trump, however, appeared to play down any hope of a major breakthrough at the Hanoi summit, saying he would be happy as long as North Korea maintained its pause on weapons testing.
"I'm not in a rush. I don't want to rush anybody," he said. "I just don't want testing. As long as there's no testing, we're happy."
North Korea conducted its last nuclear test in September 2017 and last tested an intercontinental ballistic missile in November 2017.
A South Korean presidential spokesman told reporters in Seoul on Monday that the two sides might be able to agree to a formal end of the Korean War, which ended in an armistice not a peace treaty, a move North Korea has long sought.
While a formal peace treaty may be a long way off, the two sides have discussed the possibility of a political declaration stating that the war over.
"The possibility is there," the South Korean spokesman, Kim Eui-kyeom told a briefing.
Trump is due to arrive in the Vietnamese capital, Hanoi, on Tuesday evening