In two separate readouts of the phone call, held on the eve of the Singapore meeting, the White House said Trump agreed to consult with them before his summit meeting with Kim.
"The two leaders (Trump and Abe) discussed recent developments ahead of today's summit with North Korea and agreed to consult closely following the meeting," the White House said today.
Trump spoke with Moon to discuss recent developments ahead of the North Korea summit.
"The two leaders vowed to continue their close coordination following President Trump's meeting with North Korean leader Kim Jong Un in Singapore," said a readout of the phone call.
In Singapore, Secretary of State Mike Pompeo told reporters that the US is aiming for a comprehensive, verifiable, and irreversible denuclearization on the Korean peninsula.
"We're prepared to take actions that will provide them sufficient certainty that they can be comfortable that denuclearization isn't something that ends badly for them. Indeed, just the opposite: that it leads to a brighter, better future for the North Korean people," he said.
"The United States has been fooled before -- there's no doubt about it. Many Presidents previously have signed off on pieces of paper, only to find that the North Koreans either didn't promise what we thought they had or actually reneged on their promises," Pompeo said in response to a question.
The Trump administration, he said, is going to ensure that they set up a system sufficiently robust that they are able to verify outcomes of the summit. "It's only once the 'V' happens that we'll proceed apace.
That's what's been missed before. We can go back to Reagan, 'trust but verify'," he said.
"At the end of the day, both countries are going to have to come to have sufficient trust in each other and to do the verification that each country needs that we've provided the things that are called for that we commit to in the various documents that we sign, both tomorrow, if we sign a document and if we sign subsequent documents," Pompeo said.
He added that the two countries need to ensure that they take the actions necessary to follow through on those commitments. "And when we do, we'll have a verified deal.
If we can get that far, we will have had a historic change here in Southeast Asia, North Asia, and all around the world," he added.
Pompeo said over three months, an inter-agency working group of over 100 experts across government has met multiple times a week to address technical and logistical issues associated with dismantling North Korea's weapons programmes.
"On the ground in Singapore, we have a team that includes the President's senior most expert in weapons of mass destruction who can cover any technical needs that the meetings may present.
Any suggestion that the US somehow lacks the technical expertise across government or lacks it on the ground here in Singapore is mistaken," the top American diplomat said.
North Korea, he said, has previously confirmed to the US its willingness to denuclearize.
"And we are eager to see if those words prove sincere. The fact that our two leaders are sitting down face to face is a sign of the enormous potential to accomplish something that will immensely benefit both of our peoples and the entire world," he said.
Trump believes that Kim has an unprecedented opportunity to change the trajectory of our relationship and bring peace and prosperity to his country, he said.
"We are hopeful this summit will have set the conditions for future productive talks. In light of how many flimsy agreements the United States has made in previous years, this President will ensure that no potential agreement will fail to adequately address the North Korean threat," he added.