The Sankei Shimbun and other papers reported Friday Kim had shown interest in meeting with Abe during talks with U.S. President Donald Trump.
According to the Sankei, a Japanese government official had heard from a senior U.S. government official the North Korean leader had acknowledged the "abduction issue" as a "task to be solved between North Korea and Japan."
Trump may have also brought up the issue of Japanese citizens abducted to the North during the February summit more than once.
On the first day of meetings in Vietnam, Trump allegedly told Kim at dinner there had "not been much progress" on the abduction issue. Kim did not answer, but when Trump brought up the topic again on the second day of talks, as part of "resolving serious human rights issues," Kim may have given an indirect answer.
The North Korean leader reportedly said in response a summit with Abe would be possible.
The North Korean leader's interest in a summit with the Japanese prime minister signals progress; in 2017 North Korea launched a ballistic missile that flew over Japan.
According to Asahi Shimbun on Friday, tensions at the time pushed Japan's self-defense force to consider supporting the United States militarily in the case of conflict on the Korean Peninsula.
Katsutoshi Kawano, former chief of staff of the Joint Staff Office, said responses were being considered in the event of an emergency on the peninsula.
Japan's new security laws that went into effect in 2016 were cited as reasons to take potential military action. The most recent laws allow Tokyo's self-defense forces to operate overseas.