Kang was said to have replied that the "South Korean government has worked at and will continue to work toward humanitarian issues" such as the abduction issue, as well as that of reunions for families torn apart during the 1950-53 Korean War.
Foreign Ministry spokesman Noh Kyu-duk only told reporters that Kang and Kono "will exchange opinions on matters of mutual concern".
The official Korean Central News Agency (KCNA) said in an article that there is increasing voice for re-investigation into the case of Cheonan in South Korea, as some 70 percent of South Koreans don't believe the result of investigation.
A Japanese foreign minister last visited the country in December 2015, and this marks Kono's first trip to Seoul since he took the post as Tokyo's top envoy last August.
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However, when visiting Foreign Minister Taro Kono met with his South Korean counterpart, Kang Kyung-wha, here on April 11, Kang only said, "We are not sure yet what issues will be discussed" by Moon and Kim, according to South Korean officials.
"I want Japan and South Korea to closely coordinate to realize the denuclearization of North Korea and bring about peace, stability and prosperity in Northeast Asia", Kono said at the outset of his meeting with Kang, open to the press.
During his hour-long meeting with Kang, Kono said, "No reward should be given to North Korea just because it agreed to dialogue".
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The issue of Japanese nationals abducted by North Korea remains highly critical for Japan.
"In that vein, the Ministry of Foreign Affairs is mustering a lot of efforts to maintain close communication and to strengthen cooperation with the USA through various diplomatic channels of different levels ahead of the two summits, including the South Korean Embassy in the US", the spokesman noted.
But a statement on the meeting from Seoul's foreign ministry did not mention the topic - a key issue in ties between Japan and the North, which habitually spats out angry threats of attack against "Japanese reactionaries".
The festival is held in Pyongyang as part of the commemorations for the anniversary of the birth of the North's founder Kim Il Sung on April 15, 1912.
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Kono, before the Seoul visit, also said he would try to ensure that the "abduction issue will be talked about in the North-South summit", as well as Pyongyang's "nuclear and missile issues".