Mr Trump and Mr Kim promised to work towards building "a lasting and stable peace" on the Korean peninsula and to repatriate remains of prisoners of war and those missing in action during the Korean War.
Trump initially rejected the proposal, in which North Korea would stop conducting nuclear and missile tests in exchange for a halt in military exercise between the US and South Korea.
Mr Pompeo is expected to give a more detailed account of the first-ever US-North Korea summit to Mr Moon and his South Korean counterparts, the Cheong Wa Dae official noted.
Such a meeting was unthinkable even six months ago when the two leaders found themselves in the middle of a verbal confrontation over North Korea's nuclear programme.
"I think it's very provocative".
Security experts warn that a reduction in U.S. military presence in East Asia would alter the balance of power in the region as China's engages in a rapid military build-up.
Trump also said he wants "at some point" to withdraw the American troops now in South Korea.
Max Thunder's air combat exercises so unnerved North Korea that it issued threatening statements that almost scuttled the summit.
Trump met with the North Korean leader in an unprecedented summit in Singapore, where they signed an agreement that included a North Korean commitment to full denuclearization. China, the northern neighbor of Pyongyang, said it would be willing to play a constructive role in further U.S.
It quoted Kim as saying during his talks with Trump that "in order to achieve peace and stability of the Korean Peninsula and realize its denuclearization, the two countries should commit themselves to refraining from antagonizing with each other ... and take legal and institutional steps to guarantee it". "However, Pyongyang itself, just like all of us, like Russia, China and the global community in general, has always spoken of the denuclearization of the whole Korean Peninsula".
Though the Pentagon has claimed all those war games, most of which simulate attacks on North Korea, are "necessary", analysts say they doubt suspending them will have any real impact on U.S. military capabilities.
"It will lead other dictators to think that however small a country they might run, they only need to focus on quickly building a nuclear weapons arsenal so that they can be an equal to the USA", he said.
Trump and Kim arrived in Singapore on Sunday and are set to leave later on Tuesday. -South Korean war games.
There is now confusion in Seoul and Washington DC over whether large joint exercises in August, called Ulchi Freedom Guardian, will go ahead.
"There is no single change in the suspension of the South-U.S. joint military exercise compared to the past", Nam said.
Although Trump said the process to search for more remains will begin "immediately", it's unclear when or if it will happen.
According to the Pentagon, about 28,500 USA troops are stationed in South Korea.