Mr. Trump said he was hopeful that his meeting with Mr. Kim could still take place on June 12 in Singapore, adding that resumed talks have gone very well and that the negotiations "are in good hands".
Kim Yong Chol became the first North Korea official in 18 years to visit the White House since President Bill Clinton met in the Oval Office with Jo Myong Rok, a top military official and attache to Kim Jong Il, the current leader's father.
Early Thursday, Trump told reporters "we are doing very well" with North Korea.
The historic meeting between Trump and Kim Jong-un would be the first between sitting U.S. and North Korean leaders.
The president reiterated that he had no plans to sign anything on June 12, but added that he looked forward to lifting economic sanctions now imposed on North Korea.
"We will push forward to test the proposition that we can achieve that outcome", he said.
The decision to have the meeting puts a number of Republican lawmakers in an awkward position, as many praised Trump for canceling the talks. By the next day, he was signalling the event could be back on after a conciliatory response from North Korea.
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'Come to Russia, we'll be very happy to see you, ' Lavrov told Kim in remarks released by the Russian foreign ministry.
"I haven't seen the letter yet", the president said.
Trump told reporters after the end of the meeting that he had not yet read the letter. The White House gave him a respectful greeting upon his arrival, with White House chief of staff John F. Kelly meeting him at the ceremonial diplomatic entrance on the South Lawn in full view of television cameras and photographers.
Experts cautiously raised the possibility Thursday that South Korea, North Korea and the United States could declare an end to the Korean War in Singapore following a Pyongyang-Washington summit later this month, amid preparations in full swing for the historic summit.
This handout photograph obtained courtesy of the US Department of State shows Kim Yong-chol (right), Vice Chairman of North Korea, during his dinner meeting with US Secretary of State Mike Pompeo (left) on May 30, 2018 in NY.
After trading threats of war a year ago, the two men agreed to meet for an historic summit on June 12.
Despite the upbeat messaging in the United States, Kim Jong Un, in a meeting with Russia's foreign minister on Thursday, complained about the USA trying to spread its influence in the region, a comment that may complicate the summit.
He had been personally sanctioned by the United States over his role in the North's nuclear weapons program and is thought to have masterminded an attack that sank a South Korean naval vessel in 2010, killing 46 sailors.