South Korean and US officials announced in March that Trump, despite previously trading barbs with his North Korean counterpart, had agreed to meet with Kim by May to negotiate an end to North Korea's nuclear weapons program.
Kim released the three Korean-American detainees, who had been charged with espionage or "hostile acts" against the regime, and Pompeo flew back with them to Andrews Air Force Base, outside Washington, early Thursday.
"I really think we have a very good chance of doing something very meaningful", Trump said.
Trump has previously said the release of the three men was part of working towards the denuclearization of the Korean Peninsula.
"If we can pull it off, if this president can pull it off, listen, I'll be the first person to say congratulations".
Moon is scheduled to visit the White House on May 22 to debrief Trump on his own meeting with Kim and discuss strategy ahead of Trump's summit. The family issued a statement Wednesday hailing their release. Singapore has diplomatic ties with North Korea but suspended all trade with Pyongyang in November a year ago as US-led global sanctions were tightened following a series of provocative actions by the reclusive state.
"The North Koreans have still said nothing to indicate that they are willing to give up their nuclear weapons, not to mention permanently and verifiably dismantle their nuclear programs", she said.
Mr Schumer warned Mr Trump against striking a "quick" and "bad" deal with the country.
Singapore was seen as the likely venue after Trump ruled out the DMZ on Wednesday.
During Trump's presidency, Kim has overseen a series of weapons tests to showcase his military's progress on medium- and long-range missiles and atomic weapons that put the world on edge.
The three boarded under their own steam, when most of the US delegation had already got on the plane. But the island nation was favoured by most of his advisers.
The wealthy financial and shipping hub is seen as a gateway between Asia and the West and has been called the "Switzerland of Asia", in contrast to North Korea's isolated economy that its leaders now want to modernize.
Trump is expected to meet with Kim for the first time in Singapore, though the White House has yet to reveal further details and officials cautioned that Trump may not yet have given his final approval.
The US and Singapore have a close relationship.
Technically the United States and North Korea are still at war - a stop-gap armistice ended the brutal three-year war between the two countries in 1953 and around 30,000 USA troops remain in neighboring South Korea.