The Japanese envoy also met with President Moon Jae-in, who said working together on North Korea was "more important than ever" ahead of the summits, the South's Yonhap agency reported.
Although Seoul and Tokyo present a united front in tackling North Korea's nuclear issue, their bilateral relations have frayed in recent months over historical matters, including the "comfort women" issue and Japan's claims to South Korea's eastern islets of Dokdo.
In a growing rapprochement on the Korean peninsula kickstarted by the South's Winter Olympics, Mr Kim is set to meet South Korean President Moon Jae In on April 27. But Kang stressed the importance of maintaining the momentum of "dialogue" with North Korea to achieve the goal of denuclearization, the official said.
Kono had a dinner meeting hosted by Kang, where they continued conversing about various issues including North Korea.
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.
Mitchell Reiss, director of policy planning at the U.S. State Department during the George W. Bush administration, urged caution until it is better known what Kim means when he says he is willing to talk about the denuclearization of the Korean Peninsula.
However, according to a South Korean Foreign Ministry source, Kang did not take a stance of going along with Japan's call for maximum pressure.More news: Hamilton: Ferrari dominance proof Mercedes doesn't have 'party mode'
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Moon said last month that Japan cannot unilaterally declare the wartime issue "over", while Japan says any attempt to modify or scrap the deal, signed by Moon's predecessor Park Geun-Hye, could hurt relations.
Kono emphasized that Tokyo seeks a resolution to the North Korean nuclear and missile problem and the issue of the abductions of Japanese citizens by North Korean agents in the 1970s and 1980s, according to a senior Foreign Ministry official in Seoul.
In Pyongyang's first mention of official dialogue with the U.S. and the summit with Seoul, the North's official Korean Central News Agency (KCNA) added that Kim set "strategic and tactical issues" to be upheld by the Workers' Party.
On Monday, Trump said, "I think there'll be great respect paid by both parties, and hopefully we'll be able to make a deal on the de-nuking of North Korea".
Song Tao, the head of the Chinese Communist Party Central Committee's worldwide department, will lead the mission. "The Moon and Kim summit should be getting ready for the denuclearization talks but not actually doing it for themselves".
China has sent art troupes to every festival since 1986, except in 2016.