Japan will take "full steps" to ensure the safety of its people following a North Korean missile test Tuesday that passed over the country, Prime Minister Shinzo Abe said.
May said on Wednesday Japan's upcoming trade deal with the European Union could offer a template for a future Japan-Britain trade agreement, the latest attempt to show investors that Brexit will not lead to an overnight change in business conditions.
May's first trip to Japan as premier comes amid fears among Japanese companies operating in Britain that they will lose money when it exits the EU.
"I don't really think we have to invent something really new but we will certainly discuss it with the United States", said Bessho.
"But we are obviously going to look at a whole range of different measures to put pressure on Pyongyang and we will work with our allies, with other partners, with China and Russian Federation, to see what more we can do", he said.
Pushed again if she wanted to fight the next election as leader of the Conservatives, by ITV, Mrs May said: "Yes, there's been an bad lot of speculation about my future which has no basis in it whatsoever".
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She added: "You will reap what you sow".
Officials in China, North Korea's only major ally, have called on all sides "to stick to peaceful and diplomatic means to resolve this issue".
Britain's foreign minister Boris Johnson boasted about Japan-UK investment when he visited this summer, but local firms will be looking for solid assurances from Britain's leader, who is traveling with a business delegation.
"It is my intention not just to deliver a good Brexit deal for the United Kingdom but also to ensure that "global Britain" can take its place in the world, trading around the world and that we deal with those injustices domestically that we need to do to ensure that stronger, more global but also fairer Britain for the future", she continued.
"Japanese companies are getting frustrated with the lack of answers from Mrs May on a concrete position vis a vis European Union access", Seijiro Takeshita, from the University of Shizuoka, tells the BBC.
Mr Johnson, on an official visit to Nigeria, said Mrs May was "ideally placed" to deliver Brexit and he was "here to support her".
International Trade Secretary Liam Fox is accompanying Mrs May along with 15 business leaders hoping to forge closer ties with their Japanese counterparts. "Japan and Britain will cooperate to counter this". Roughly 1,000 Japanese companies employing around 140,000 people are now operating in the UK.