Moon is only the third South Korean leader to visit Pyongyang since the end of World War II.
The book outlines how Trump told former staff secretary Rob Porter that the tweet "may be my best ever."
"China needs to kill him and replace him with a North Korean general they control," the senator apparently said.
The Mass Games is a huge, nationalist pageant performed by up to 100,000 people in what is believed to be the world's largest stadium.
"We will both prove everyone wrong!" Trump said of himself and Kim Jong Un on Twitter. "There is nothing like good dialogue from two people that like each other! Much better than before I took office."
"If both sides have the political willingness, it is reasonable for them to end the technical state of war on the peninsula first."
"The safety and security of U.S. citizens overseas is one of our highest priorities," a State Department official said.
The International Atomic Energy Agency called the "further development" of Pyongyang's nuclear program a "cause for grave concern" in a report.
The North Korean regime's official news agency said Putin told Kim he would like to meet him soon.
Once DNA sampling begins, the process could take six months to obtain the first results. A number of other obstacles will likely slow the process.
Skepticism over North Korea's intentions mounts.
Recent satellite images suggest North Korea may finally be moving to dismantle a test site but doubts and unclarity remain over how Kim intends to implement "denuclearization."
The president struck back at the story in a tweet Monday.
South Korea said the move "will contribute substantially to the alleviation of military tensions and the building of trust between the two Koreas."
Pyongyang said it would "protect the independent and creative life of the popular masses and make selfless, devoted efforts for them."
North Korea needs oil products to develop ballistic missiles and nuclear weapons, according to experts.
President Donald Trump tweeted the letter to him from North Korean supreme leader Kim Jong Un, who shared kinds words but little policy change.
Veteran U.S. diplomat Robert Gallucci said Trump's "tweeting on the subject of war and peace with North Korea isn't the most useful thing in the world."
"Contrary to our military's suspension of joint Korea-U.S. military exercises, North Korea is conducting military exercises on a scale similar to previous years."
KCNA Watch has registered the lowest levels on its Threat Index since North Korean supreme leader Kim Jong Un came to power in 2011.
This year North Korea recognizes July Fourth differently.