Mr Kim, a close aide to the North Korean leader, is the highest level figure from the secretive state to hold talks at the White House since a senior envoy visited former president Bill Clinton in 2000. "Hopefully we will have a meeting on the (June) 12th", he said.
The historic meeting between Mr Trump and Kim would be the first between sitting United States and North Korean leaders.
"We'll be meeting in June 12 in Singapore", Trump said, after Kim Yong Chol drove away.
Trump later told the Reuters news agency it may take more than one meeting with Kim Jong Un to seal a denuclearization deal with North Korea and that he would like Pyongyang to dismantle its nuclear weapons program as quickly as possible under any agreement.
Russian Foreign Minister Sergei Lavrov met North Korean leader Kim Jong Un in Pyongyang on Thursday and invited him to visit, moving to raise Moscow's profile in global efforts to ease tensions on the Korean peninsula.
The Trump told them "it was a very interesting letter".
Now, diplomatic proclamations, like letters of credence for ambassadors, often come in a nice folder, because you don't want to fold those. How much? How much?
He also said that North Korea "wanted to do denuclearisation", although the North did not confirm this.More news: Trump escalates fight with ABC after 'Roseanne' cancellation
As the two were meeting, so were lower-level USA and North Korean diplomats - both in Singapore, the planned venue for the historic summit, and at the Koreas border. Hours later, Trump said the summit was potentially back on.
"We'll be meeting on June 12 in Singapore", Trump told reporters, noting that the meeting at the White House today "went very well".
Song Young-moo says, "Just because we have been tricked by North Korea in the past doesn't guarantee that we will be tricked in the future".
His travel restrictions were lifted this week so he could fly to NY and Washington, DC to meet U.S. officials, the Department of State said.
Trump had previously cancelled the summit on May 24, citing a "recent statement" by North Korea as the cause for the cancellation.
He is also linked to the cyber-attack on Sony Pictures in 2014 and led the country's delegation to the 2018 Winter Olympics closing ceremony in South Korea's Pyeongchang earlier this year. Marco Rubio (R-Fla.) tweeted after Trump announced the cancellation of the meeting.
The North Korean leader laid his position out during a visit by the Russian foreign minister, Sergei Lavrov, on Thursday.
United States officials with knowledge of the letter's contents said it did not make any noteworthy threats or concessions, and was seen as a positive step, according to multiple news reports.