Sen. Josh Hawley (R-Mo.) is pressing the Federal Trade Commission to inform Congress about any gaps in the agency’s ability to hold tech giants such as Facebook Inc. and Alphabet Inc.’s Google accountable to U.S. antitrust laws.

In a March 11 letter to the FTC, Hawley said the agency’s current enforcement approach has been “toothless,” failing to stop big-tech companies from harming consumers or crippling emerging rivals. He indicated that he was open to working with the FTC to address the matter.

The letter by the freshman lawmaker who serves on the Senate Judiciary Committee’s antitrust panel is the latest...