When Ford Motor Co. Executive Chairman Bill Ford Jr. called the president last spring, he was hoping to defuse what was looking like a lengthy legal battle over the nation’s fuel-economy regulations for vehicles.

The Trump administration wanted to significantly ease U.S. environmental rules. California had sued to stop that. The car industry was caught in the middle. Mr. Ford, calling from his Dearborn, Mich., office, urged the president to broker a compromise with California, according to people with knowledge of the conversation.

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