Any such visit would be controversial, especially in Seoul and Washington, which have led the charge to further isolate Pyongyang over its ballistic missile and nuclear programmes.
Tokyo and Pyongyang have no formal diplomatic ties, partially because of what Japan says is the North's unwillingness to come clean over the abductions in the 1970s and 1980s.
But in a breakthrough last week, they said an investigation into the fate of missing Japanese would be re-opened. In exchange, Japan would ease some of the unilateral sanctions it has imposed on the isolated state.
"In doing so, we will consider (Abe's) making a visit to North Korea," he said, according to Jiji Press news agency.
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Kishida noted that the government needed to act swiftly as families of kidnap victims are increasingly elderly, but said that nothing had been decided about a possible prime ministerial visit yet.
Abe's right-hand man, Chief Cabinet Secretary Yoshihide Suga, said at the weekend that the government would send officials to North Korea to monitor the probe.
Five of them returned home but Pyongyang said without producing credible evidence that eight of them had died, provoking an uproar in Japan.
There are suspicions in Japan that dozens - perhaps hundreds - of people were taken.
The abductions issue is a highly emotive one that colours all of Japan's dealings with North Korea.
However, the international community, led by Washington, is primarily focused on ridding the unpredictable regime of its ballistic missiles and its nuclear programme.