"I hope the North Koreans will come to the table with ideas that they didn't have the first time. We hope we can we be a little more creative too," says U.S. Secretary of State Mike Pompeo.
North Korea released and deported an Australian student after a week of detention for alleged espionage and spying.
North Korea's mission to the United Nations has accused the U.S. of being "obsessed with sanctions" against Pyongyang.
"The president's people are shocked," says CNBC Jim Cramer.
"Persuading countries not to go nuclear is hard," ex-diplomat John Negroponte, said.
While the two spoke of reconciliation, Trump said that U.S. sanctions on the country over its nuclear weapons and missile development programs would stay for now.
Before setting off for Asia, Trump sent a letter to Kim. The White House confirmed correspondence between the two leaders was ongoing.
News of the letter comes as Trump prepares to depart for Asia, where he is expected to hold talks with Chinese President Xi Jinping.
North Korean leader Kim Jong-un and Chinese President Xi Jinping agreed that amid "serious and complicated" international affairs strengthening bilateral ties was good for both countries and regional peace.
China's President Xi Jinping arrived in Pyongyang on Thursday morning for a state visit to North Korea — the first by a Chinese state leader in 14 years.
President Xi Jinping's visit to North Korea this week will mark the first time in 14 years that a Chinese leader is visiting the reclusive country.
Chinese President Xi Jinping said in an op-ed in North Korean state newspaper Rodong Sinmun on Wednesday that China supports North Korea's "correct direction" in resolving the issue of the Korean Peninsula politically.
China's Xi Jinping will travel to North Korea this week, ahead of a possible meeting with President Donald Trump at the G-20 summit in Japan.
Chinese President Xi Jinping will make a state visit to North Korea this week, state media announced Monday.
North Korea's Rodong Sinmun newspaper quoted the country's leader, Kim Jong Un, as saying "prevention is the key to production in livestock industry".
North Korean senior official and former top nuclear envoy Kim Yong Chol reportedly accompanied leader Kim Jong Un to a Sunday art performance signaling that the former spymaster is alive and remains a force in North Korea's power structure.
With the stock market already on track for its first negative May since 2012, the final day of the month begins with more sharp losses.
North Korea executed Kim Hyok Chol, its special envoy to the United States, and foreign ministry officials who carried out working-level negotiations for the second U.S.-North Korea summit in February, a South Korean newspaper reported on Friday.
North Korea asked United Nations Secretary-General Antonio Guterres to deal with the "illegal" seizure of one of its cargo ships by the United States, state media said Saturday.