TOKYO—Japan’s biggest lender, Mitsubishi UFJ Financial Group Inc., is stepping into the bitcoin world by investing in Coinbase, a service that allows users to store the virtual currency and make payments with it.

To ramp up operations in Asia, Coinbase is set to raise about $10.5 million from investors on Friday including MUFG’s core banking unit and an MUFG venture-capital fund, bank officials said.

The...