Francis told visiting South Korean President Moon Jae-in he could travel to the isolated state if Kim sent him an official invitation, South Korean news service News 1 reported.
An official invitation from Kim would be met with an unconditional response, the pope said.
Yoon Young-chan, the president's chief press secretary, relayed the pope's conversation with Moon to the South Korean press pool.
Moon has played a mediating role in engagement with the North, and on Thursday he delivered the message from North Korea.
"Last month when I visited Pyongyang I told Chairman Kim your Holiness takes great interest in peace and prosperity on the Korean Peninsula," the South Korean leader said. "At that very [moment] Chairman Kim said he would warmly welcome your Holiness' visit to Pyongyang."
Moon also said Kim "expressed gratitude" to the pope for his prayers for peace on the peninsula.
Francis did not say yes to the invitation from Moon, saying instead he will give an answer upon receiving a formal North Korean invitation.
The pope said he strongly supports the South Korean government's efforts to promote peace between the two Koreas.
Francis also told Moon to "not stop and move forward, without fear," South Korean news service Money Today reported.
Moon said the past advice from Francis encouraged him to pursue dialogue with the North.
"At every moment of difficulty in the past year, your Holiness' advice, that only dialogue can resolve all conflicts, was deeply carved in my heart," Moon said.
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