TOKYO—Japan’s biggest banks have never seen a need to connect their automatic teller machines to global networks, leaving many foreign visitors frustrated and short of cash.

Now the country’s three biggest lenders—Mitsubishi UFJ Financial Group Inc., Sumitomo Mitsui Financial Group Inc. and Mizuho Financial Group Inc.—are relenting somewhat after several years of pressure from a government looking to build on a tourism boom ahead of the 2020 Olympic Games in Tokyo.

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