In his speech to the 74th United Nations General Assembly late Tuesday, Abe, the only world leader in Northeast Asia who has yet to meet with Kim, offered to meet the North Korean leader without preconditions.
"Japan's unchanging objective is to normalize relations with North Korea through comprehensively resolving the outstanding issues including the abductions, nuclear and missile issues," Abe said. "I am determined to meet with Chairman Kim Jong Un myself without attaching any conditions."
Abe has previously proposed a summit with Pyongyang at the U.N.
Last year, the Japanese prime minister called for the resolution of the abduction issue, which could be resolved through a "new start," or a direct meeting with Kim.
The two sides have made little progress in negotiations, despite some lower-level meetings, regarding abducted Japanese, in Mongolia in 2018.
Last week, North Korea condemned Japan, following Japanese news reports linking recent discoveries of dead Japanese in Japan to North Korea kidnappings. Pyongyang has also accused Japan of politicizing and internationalizing the issue of missing Japanese -- individuals who disappeared in the late '70s to early '80s.
Tokyo still recognizes a dozen abductees, although Pyongyang has said none are alive.
North Korea's lack of cooperation on the issue comes at a time when it has returned to tests of weapons systems. The Kim regime has also turned down South Korean calls for cooperation while remaining open to restarting working-level talks with the United States.
On Tuesday Abe endorsed U.S. President Donald Trump's North Korea policy, which Trump defended before the U.N. earlier in the day as "bold diplomacy."
"Japan supports the approach taken by President Trump," Abe said Tuesday.
The Japanese prime minister also said the "approach in which the two leaders [Trump and Kim] talk candidly with each other" could lead to a "bright future."
"It has changed the dynamics surrounding North Korea," Abe said.
As the prime minister seeks to ease tensions with North Korea, his relations with the South has become more strained, following his decision to place trade restrictions impacting South Korean tech companies.
Abe did not address ties with Seoul, an area of concern for the United States as Washington seeks trilateral cooperation among partners to deter North Korea and China.
Earlier on Tuesday the U.S. House of Representatives passed a resolution emphasizing the importance of U.S. alliances with South Korea and Japan, according to South Korean news agency Yonhap.
Abe instead turned his attention to the Middle East, saying Japan "shares concerns" regarding the recent attack on Saudi Arabia's crude oil facilities, while describing the incident as a "contemptible crime" that holds the international community "hostage."
Abe suggested he is mediating tensions between the United States and Iran, and that Iran has said it is committed to not producing, possessing or using nuclear weapons.
Trump had said earlier on Tuesday the "repressive regime" in Iran poses the "greatest threat" to the United States.
"I will never fail to defend America's interests," Trump had said.