Pompeo, who was director of the Central Intelligence Agency before becoming secretary of state in April when Trump fired Rex Tillerson, has met twice with Kim in Pyongyang.
If it goes ahead, it will be first meeting between a US and a North Korean leader during more than six decades of hostility, and it would come just months after the North's rapid progress toward attaining a nuclear-tipped missile that could strike America fueled fears of war.
"That decision will ultimately be up to Chairman Kim", he said.
He did not detail the conditions he had laid out for Kim but said if they aren't met, "we won't have the meeting".
Trump raised doubts about the summit in talks on Tuesday with South Korean President Moon Jae-in, who came to Washington to urge Trump not to let a rare opportunity with reclusive North Korea slip away.
"Our posture will not change until we see credible steps taken toward the complete, verifiable and irreversible denuclearisation of the Korean peninsula".
"A bad deal is not an option".
Pompeo said that "if we can get America's interests safe and secure we are prepared to do a great deal", including security assurances for Kim. "It could very well happen", Trump said of the on-again-off-again June 12 meeting, which has been as keenly teased as any of his "The Apprentice" season finales. If the right deal is not on the table, we will respectfully walk away.
Mr Pompeo has twice met Mr Kim in Pyongyang. Press Secretary Sarah Sanders said Trump was referring to a commitment to seriously discuss denuclearization.
But Kim "has shared candidly that he understands that economic growth for his people, the well-being of his people, depends on a strategic shift", he said.
Sam Allardyce expected to leave Everton on Wednesday
Appearing on Sky Sports on Monday, Allardyce admitted that he did not know what his future would hold beyond the meeting with Moshiri.
Previous similar gestures by the North were rapidly reversed when the worldwide mood soured.
North Korea had earlier refused to grant entry visas to the South Korean journalists after the North cut off high-level contact with Seoul to protest joint U.S.
Pyongyang then accused US National Security Adviser John Bolton of making "reckless statements" for his suggestion that the North could follow a "Libya model" of denuclearisation.
The US secretary of state was speaking in Congress a day after Trump had cast doubt on the timing of the summit in Singapore, and appeared to drop the insistence of "all-in-one" unilateral disarmament before receiving any reciprocal steps from the US. Libya's longtime autocratic leader Moammar Gadhafi was killed several years later after a Western-backed military intervention. Trump indicated he believes the meeting will take place, but left open the possibility it would be delayed or even cancelled if a fruitful outcome doesn't seem likely.
Politically, Trump has invested heavily in the success of the planned summit, leading most United States officials amd outside observers to believe it will go ahead.
Mr Trump later denied the USA would follow the "Libyan model" if an agreement was reached with North Korea.
For starters, there'd be the craved photo of him shaking hands with a sitting USA president.
Mr Trump accepted North Korea's invitation for direct talks after more than a year of heated rhetoric and with global concern that hostilities might escalate into military confrontation.
"North Korea can say to the worldwide community that it did its best to achieve denuclearization through negotiations but was pressured by the United States and couldn't do it", he said.
And the Post reported Trump reassured Kim of his safety.