"During the summit with Trump, Kim told Trump 'I can meet with Prime Minister Shinzo Abe'," the Sankei reported.
Japan wants the talks to push the emotive issue of citizens abducted by the North decades ago, which has seen little movement despite a whirlwind of diplomacy in recent months.
Abe today repeated a pledge to push for dialogue with Pyongyang on the issue as he met families of abductees.
"I will face (North Korea) directly and work toward resolving the abduction issue," he told the families.
"Japan has to take the initiative to resolve the issue," he said, adding that the summit would be "meaningless if it yields no progress on the abduction issue".
Another scenario would see Abe meet Kim on the sidelines of a conference in Russia in September, the daily said.
Several Japanese media outlets said Kim had expressed a readiness to meet Abe when he held a summit with Trump. And Abe has already said publicly that he would be willing to meet Kim in order to resolve the abduction issue.
"If (Abe's) visit to Pyongyang in August proves difficult," he could hold talks with Kim on the sidelines of the Eastern Economic Forum in Russia's Vladivostok in September, the Yomiuri said.
Japanese media, including Jiji Press, said Japanese foreign ministry officials "contacted" their North Korean counterparts Thursday at an international security conference in Mongolia.
The issue of Japanese citizens who were abducted in the 1970s and 1980s to help Pyongyang train its spies has long soured already strained relations between Tokyo and Pyongyang.
The Japanese government has officially listed 17 people as abductees, but suspects dozens more were snatched.
Sakie Yokota, whose daughter Megumi was kidnapped at age 13, told reporters she was encouraged by Trump's talks with Kim and said she had urged Abe to resolve the issue quickly.
"This is the beginning of a beginning," added Shigeo Iizuka, whose sister was kidnapped four decades ago.
"I can only say 'I'm sorry' to my sister... But I want to tell her, 'Don't give up, hold out a bit more,'" he added.
Trump said Tuesday he discussed the abductee issue with Kim, but it was not mentioned in the document signed by the two leaders.
Japan has maintained a hardline position on North Korea despite the stepped-up diplomacy with Pyongyang in recent months, and has been left largely on the sidelines as South Korea, China and the United States have held talks with Kim.
(This story has not been edited by Business Standard staff and is auto-generated from a syndicated feed.)