His 12-day tour includes stops in Japan, South Korea, China, Vietnam and the Philippines.
President Donald Trump has said he can not understand why Japan didn't shoot down North Korean missiles as they flew over the country earlier this year. Ahead of the dinner party, Trump said the "relationship is really extraordinary". "I have no doubt that it will be done in a quick and very friendly manner".
In his public appearances so far, Trump has sought to cast himself as a highly successful chief executive, touting the United States economy while taking credit for a streak of hiring by American companies. USA officials have said privately that intercepting a test missile is among options under consideration, though there is disagreement within the administration about the risks.
Trump all but pleaded with Japanese carmakers to build more factories in the US, or let a few more American cars into Japan.
Donald Trump and his Japanese counterpart, Shinzo Abe, started by delicately spooning out the food into the pond to the waiting koi, which had been rounded up by a clapping Japanese aide.
Renegotiating South Korea's bilateral free trade agreement with the USA, dubbed KORUS, is likely be high on the agenda during the Trump-Moon summit.
The two sides remain at odds over how to frame future trade talks, with Tokyo pushing back against USA calls to discuss a bilateral free trade agreement (FTA).
The US President described North Korea's nuclear programme as "a threat to the civilised world and global peace and stability" on the second day of his Asia tour dominated by the crisis. Trump also said earlier that an Indo-Pacific trade framework would produce more in trade that the Trans-Pacific Partnership pact pushed by his predecessor but which he announced Washington would abandon soon after he took office.
While it was not the first time Trump had signaled his openness to meeting with Un, he and his administration have continued maintain that while they aren't ready to close he door on talks, now is not the right time. After decades of official silence, Pyongyang finally admitted in 2002 that it had operated a state-sponsored program to kidnap Japanese citizens and bring them to North Korea to teach Japanese language and culture for purposes of espionage. US producers are concerned they're losing market share, and would like reduced tariffs on beef, pork, dairy, fruit and vegetables, among other foods. Trump met Emperor Akihito, exchanging a handshake and nodding, before his lunch and talks with Abe.
A total of 18 North Korean bankers stationed in China, Russia and Libya with suspected links to the regime's weapons programmes have been blacklisted, a statement posted on the South's government website showed.
Abe thanked the president for referring to the abduction issue in his September speech to the United Nations General Assembly. He ominously added that, "Every once in a while, in the past, they underestimated us".
Trump on Monday talked up recent Japanese purchases of defensive military equipment created to blunt the threat of North Korea's nuclear program.
Abe grinned slightly, but it was an awkward moment after a trip in which the two men went out of their way to call each other friends and stress the bonds between the two allies, which both men said were the strongest ever.
Though Trump and Abe repeatedly touted their friendship, looming disagreements on trade could strain the friendship.