Trump said he would take a new approach to Kim Jong Un. "I would speak to him," he said.
Pyongyang is believed to be "four to five years" away from reaching the U.S. mainland, 5,700 miles away.
The country could now be capable of hitting the U.S. mainland.
The two men met at the Asian Leaders Conference in the South Korean capital.
North Korea has tested ten missiles in 2017 and could be close to developing a nuclear warhead.
Trump and South Korea's Moon meet in the shadow of the death of Otto Warmbier.
Pyongyang is ramping up anti-American fervor ahead of the "day of struggle against U.S. imperialism" on June 25.
The 22-year-old University of Virginia student was sentenced to 15 years' hard labor.
The former NBA star has called the North Korean leader a 'friend for life.'
The country has carried out five nuclear tests in defiance of U.N. Security Council resolutions.
The president doesn't have independent constitutional authority to start a war with a foreign nation.
The failed attempt happened just hours before U.S. Vice President Mike Pence was due in South Korea for talks on the North's increasingly defiant arms program.
Earlier speculation suggested it may have had something to do with marking the birthday of the deceased founder of the country.
North Korea celebrates its founder's birthday on Saturday and has tested missiles on significant occasions in the past.
President Donald Trump has sent a Naval strike group toward the Korean Peninsula as the threat of a North Korean nuclear test looms.
North Korea's Asian Cup tie with Malaysia has been postponed.
China said that international action should wait until the investigation is complete.
A man claiming to be Kim Han Sol released a video saying that he is in hiding with his mother and sister.
Malaysian police investigating the murder of Kim Jong Nam questioned up to three men hiding in the North Korean embassy.
Regional stakes high given North Korea's threat to test an intercontinental ballistic missile.