In his talks with Mr Trump in Argentina, Mr Moon earned some support for the Seoul summit from the USA leader - perhaps in a bid to entice Kim to make good on his pledge.
The president tells reporters that there are three sites under consideration, but he's not identifying them.
The Koreas reached the military agreement during the third summit between Kim and South Korean President Moon Jae-in as part of a diplomatic offensive that began earlier this year, reversing months of rising tensions that had threatened to erupt in a nuclear war.
As officials work to arrange a second meeting between Trump and Kim, South Korea's Moon Jae-in relayed a message to reporters given to him by the United States president to pass on to the North's leader.
"We're getting along very well", Trump told reporters aboard Air Force One during the return trip from the G20 summit in Buenos Aires, Argentina.
South Korea sent rail cars and dozens of officials to North Korea on Friday for joint surveys on northern railway sections the countries hope someday to connect with the South.
Trump and Xi agreed that "great progress has been made with respect to North Korea", the White House statement added.
The incident did not trigger any gunfire, unlike a year ago when a North Korean soldier ran across under a hail of bullets from his own side.
President opens up on North Korea, Pakistan, Turkey and Saudi Arabia. But critics say the South risks conceding some of its conventional military strength before North Korea takes any meaningful steps on denuclearization, as the larger nuclear negotiations between Washington and Pyongyang seemingly drift into a stalemate.
The US has urged North Korea to take concrete steps towards denuclearization, whereas Pyongyang has demanded guarantees for its survival before the signing of a peace treaty.
But differences remain between Washington and Seoul on how to proceed with Kim, as the dovish Moon favours more robust engagement with the North.
Prospects for a second summit to overcome a current logjam in negotiations will likely depend on whether either the USA or DPRK can show flexibility with regards to offering gestures surrounding sanctions relief or steps towards denuclearization.