The pontiff made a five-day trip to South Korea in August 2014, his first visit to Asia as pope, during which he called for peace and reconciliation on the Korean peninsula.
Tensions are rising: USA and Chinese destroyers had a near-miss in the South China Sea recently near Gaven Reef, part of the disputed Spratly archipelago where China has developed military outposts.
"We will make - we will figure it out and make it work", Pompeo told his greeter. Chairman Kim said he's ready to allow them to come in.
South Korean President Moon Jae-in said that Kim is expected to visit Russian Federation soon and that Chinese leader Xi Jinping will also travel to North Korea, but did not give any further details.
China and the United States are at odds over trade and Beijing's militarization of islands it claims in the South China Sea.
Trump emphasised the rapport he says he's developed with the North Korean leader as they negotiate over United States demands that Kim abandon his country's nuclear weapons and ballistic missile programmes.
The North's regime since then has allowed Catholic organisations to run aid projects in the impoverished country, but direct relations with the Vatican are non-existent.
North Korea's vice foreign minister Choe Son Hui met with officials in Beijing on Saturday and is due to hold talks with Russian counterparts in Moscow followed by a trilateral meeting with China and Russia.
During Pompeo's meeting in Pyongyang, Kim also invited inspectors to visit the Punggye-ri nuclear test site to "confirm that it has been irreversibly dismantled", the State Department said.
Rodong Sinmun, North Korea's state newspaper, featured eight photos from the meeting.
Saying the two sides' cooperation should be supported by healthy and steady bilateral ties, Wang expressed his hope that Pompeo's Beijing visit will play a positive role in realizing the goal.
Trump "will not back down" from the challenge, Pence said.
"It's a very nice day that promises a good future", Kim said, "for both countries".
The secretary of state said the two sides were "pretty close" to deciding on a time and place for the second summit, declining to provide details.
Pope Francis has made repeated public appeals for peace on the Korean peninsula, and led the faithful gathered in St. Peter's Square in prayers for the U.S.
Rising tensions have fueled concern in Beijing and Washington that the president's trade fight could, if left unchecked, foster a new Cold War between the two countries.
It added that there was an agreement to hold working-level talks to arrange a second summit between Kim and Trump.
Mr Pompeo described his fourth trip to the North Korean capital as "another step forward to denuclearisation".