PALO ALTO, Calif.—Roughly a year ago, Google offered health-data company Cerner Corp. an unusually rich proposal.

Cerner was interviewing Silicon Valley giants to pick a storage provider for 250 million health records, one of the largest collections of U.S. patient data. Google dispatched former chief executive Eric Schmidt to personally pitch Cerner over several phone calls and offered around $250 million in discounts and incentives, people familiar with the matter say.

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