Speaking at the end of a southeast Asian tour, Abe called for reconciliation after a year of diplomatic discord marked by a territorial dispute and visits by Japanese politicians to a controversial war shrine that angered South Korea, its former colony.
"With President Park Geun-Hye I had a telephone conversation, we belong to the same generation so I do sincerely hope that we will be able to hold a summit meeting," he told reporters.
He did not say when the phone call took place nor when the summit, which would be the first between the two leaders since they came to power, could occur.
However he described Seoul as "a most important neighbour with whom Japan shares fundamental values and interests".
"Currently communications are being made between the foreign affairs authorities but in a a calm and tranquil atmosphere. Dialogue should be held so that the bilateral relationship can steadily progress," Abe said through an interpreter.
Abe's comments came after South Korea's First Vice Foreign Minister Kim Kyou Hyun met with Japanese Foreign Minister Fumio Kishida in Tokyo the previous week.
Despite common concerns over North Korea's weapons programme, Japan and South Korea have quarrelled, often over issues stemming from Japan's wartime aggression in Korea.
There are longstanding issues as in demands for reparations for Korean "comfort women" who were forced to provide sex to Japanese soldiers during World War II.
There was also a disagreement last year over the ownership of two sparsely populated islands which Seoul controls and refers to as the Dokdo but which Tokyo claims as Takeshima.
There have also been concerns about Abe's plans to revise Japan's constitution, removing pacifist provisions that have restrained Japan's military.
However Abe said a greater Japanese presence would have a positive effect.
"Japan, together with the United States, has been greatly contributing to the stability of the Asia-Pacific region in going forward and we intend to continue playing that role," he said.
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