Bank of America Corp. is working on sweeping changes that would require many users of basic checking accounts to pay a monthly fee unless they agree to bank online, buy more products or maintain certain balances.

The plan by the nation's second-largest bank by assets is the latest sign of stresses in the banking industry at a time of low interest rates, slow economic growth and new rules limiting many types of service charges. Many other big banks, including J.P. Morgan Chase & Co.—the nation's largest—and Wells Fargo & Co.,...