The Justice Department is investigating how the FBI handled sexual-abuse allegations against former U.S. gymnastics national-team doctor Larry Nassar, amid claims agents failed to respond to complaints from gymnasts in 2015, people familiar with the matter said.

The move by the department’s Inspector General’s office follows an internal Federal Bureau of Investigation review opened earlier this year into the bureau’s handling of the Nassar allegations. The gymnasts’ complaints languished for at least nine months before an...