"This is the first time I have met someone with the same kind of guts," or fire in the belly, Kim reportedly said upon meeting Pompeo.
Kim also expressed "joy" and "satisfaction" upon meeting the U.S. spy chief known for his tough position on North Korea's nuclear weapons program.
After the trip was disclosed to the public last week, a U.S. official told the Asahi the North Korean leader had conveyed an intention to denuclearize in negotiations with the United States.
Pompeo did not travel alone -- he was part of a six-member U.S. delegation to North Korea.
The meeting likely took place over Easter weekend, according to the Asahi, and Pompeo exchanged views with Kim in about three to four separate meetings during his three-day, two-night stay in North Korea.
Kim expressed his intention to give up his nuclear weapons program while not asking for the withdrawal of U.S. forces.
He also said he intends to free the three remaining U.S. detainees in North Korea.
Yonhap reported Monday that Kim's landmark summit with South Korean President Moon Jae-in will entail his stepping into South Korean territory, but Kim will be exempt from documentation requirements.
The exemption overrides South Korean law that stipulates citizens on both sides must receive approval from the unification ministry prior to visiting the other side of the DMZ.
Local news service Financial News reported Monday Kim will be given the highest welcome by Moon that includes a ceremony, a state banquet and other gestures reserved for top guests.
Seoul has yet to confirm whether North Korea's first lady, Ri Sol Ju, will be visiting, and has not ruled out the possibility, according to the report.