Pompeo is holding a joint news conference with South Korean Foreign Minister Kang Kyung-wha.
Pompeo has had two recent meetings with Kim to prepare for the summit and, last weekend, to negotiate the release of three Korean-Americans held in the North's jails.
North Korea has long seen its nuclear program as key to its survival, and it's not certain how honest Kim Jong-un is about giving up his nuclear arsenal.
Ms Kang praised the upcoming meeting between Mr Trump and Mr Kim in Singapore as a "historic" opportunity, but added a few notes of skepticism as well.
Pompeo didn't go into details, but he suggested the US could help the North Korean economy if it were no longer a nuclear threat. Therefore, she said, any change in the size of the USA forces in the South Korea should not be on the table at the summit.
Abe also said global sanctions should remain in place against North Korea and any rewards must wait until Kim fully abandons his nuclear and missile programs. "Conversations that involved deep complex problems, challenges; the strategic decision that Chairman Kim has before him about how it is he wishes to proceed and if he is prepared, in exchange for the assurances we are ready to provide to him, if he is prepared to fully denuclearise".More news: CLE 128, TOR 93: James, Cavs sweep Raptors to make conference finals
"We'll require a robust verification program, one that we would undertake with partners around the world which would achieve that outcome", he warned.
"If Chairman Kim chooses the right path, there is a future brimming with peace and prosperity for the North Korean people", he said.
"There's a danger here of the peace track moving more quickly than the denuclearization track", warned Abraham Denmark, an Asia expert and former senior U.S. defense official.
He described his conversations with Kim as "warm" and said both had tried "our best to make sure that we were communicating clearly and had shared understanding about what our mutual objectives were".
However, his comments made clear that the two sides remained far apart on the key issue of what they mean by denuclearization.
After her talks with Pompeo, she took pains to "emphasize again that the US military presence in Korea is a matter for the ROK-US alliance first and foremost".
To the United States, that means the North giving up the nuclear weapons it has already built.