Mr Kim arrived in town on Sunday for a historic summit on Tuesday with US President Donald Trump to discuss possible denuclearisation efforts.
"The ultimate objective we seek from diplomacy with North Korea has not changed - the complete, verifiable and irreversible de-nuclearization of the Korea peninsula is only outcome that the United States will accept", declared Pompeo.
Mr Kim - who is the Chairman of the State Affairs Commission, North Korea's highest decision-making body - also shook hands with Dr Vivian, Mr Ong, Senior Minister of State for Foreign Affairs as well as Defence Maliki Osman and Senior Minister of State for Communications and Information as well as Culture, Community and Youth Sim Ann.
The two men spoke privately before a backdrop of red, white and blue flags, North Korea's single star interspersed with the stars and stripes of the American banner.
In Trump's and Kim's meeting on Tuesday, reporters were allowed access for just a few minutes at the beginning. Kim said, "It has not been easy to get here" and that "the old prejudices and practices worked as obstacles, but we have overcome them and we are here today".
Diplomats and foreign policy experts have criticized the idea of a USA president sitting down solo with the North Korean dictator, saying it heightens the potential for a misleading readout from Kim that could come into conflict with Trump's account of the exchange.
Trump called the prime minister a friend and said "this was a choice that we made very consciously".
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Trump was hosted for a working lunch Monday by Singapore's prime minister, Lee Hsien Loong, at the Istana, which is the official residence of the city state's president.
The country is also no stranger to hosting historic summits, having in 2015 been the site of a meeting between Chinese President Xi Jinping met Taiwan President Ma Ying-jeou - the first time leaders from both sides met since Taiwan's separation from the mainland in 1949.
Trump has insisted that Pyongyang must give up its nuclear weapons ambition. "And it's my honour and we will have a terrific relationship".
"I think that I brought awareness to a lot of things around the world and I think North Korea has given a lot of people the opportunity to do this conference now and I hope it's a success".
"For long-time Korea-watchers, seeing the first-ever handshake and meeting between a sitting US president and North Korean leader is already momentous". The U.S. president turns 72 on June 14.
"Thank you for also asking to hold the summit in Singapore", added Mr Lee.
Still, he said, Mr. Trump's aim with Mr. Kim was clear: to build trust by creating a personal bond.