The Federal Bureau of Investigation is probing a computer-hacking attack on J.P. Morgan Chase & Co. and as many as four other banks, in what people familiar with the probe described as a significant breach of corporate computer security. (Further reading: J.P. Morgan says about 76 million households affected by cyber breach).

The timing and extent of the hacking attacks weren't immediately clear, though cybersecurity experts began probing the possible J.P. Morgan breach earlier this month, according to people familiar...