The Justice Department wants the U.S. Supreme Court to address whether Google users suffered enough injury to bring privacy claims that resulted in an $8.5 million class settlement.

That’s not the issue the top court agreed to hear when it accepted Frank v. Gaos for review in April. The petition for review asked whether the settlement was fair, given that all the money goes to plaintiffs’ attorneys and charities that work on privacy-related issues.

But if the justices do take up the injury question, big business could get the victory it sought before Justice Antonin Scalia died in 2016.

Justice...