An effort by the FBI to more aggressively monitor social media for threats sets up a clash with Facebook Inc.’s privacy policies and possibly its attempts to comply with a record $5 billion settlement with the U.S. government reached last month.

The Federal Bureau of Investigation is soliciting proposals from outside vendors for a contract to pull vast quantities of public data from Facebook, Twitter Inc. and other social media “to proactively identify and reactively monitor threats to the United States and its interests.”...