The surprise disclosure that agents from the Federal Bureau of Investigation are taking a new look at Hillary Clinton’s email use lays bare, just days before the election, tensions inside the bureau and the Justice Department over how to investigate the Democratic presidential nominee.

Investigators found 650,000 emails on a laptop that they believe was used by former Rep. Anthony Weiner and his estranged wife Huma Abedin, a close Clinton aide, and underlying metadata suggests thousands of those messages could have been sent...