WASHINGTON—Phone companies would shoulder much of the workload under a revamped version of the National Security Agency's U.S. phone-data surveillance program, as detailed Thursday by the White House.

Under the proposal—much of which was revealed earlier this week—the government would no longer store phone records in bulk. Instead, those would remain at telecom companies, to be searched when officials obtain a court order or, in emergencies, without such an order.

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