Exxon Mobil Corp. Chief Executive Rex Tillerson, who was meeting with Donald Trump on Tuesday to discuss becoming his secretary of state, is a seasoned deal-maker whose close ties to Vladimir Putin and other world leaders could redefine American interests abroad.

His emergence as a candidate to be the nation’s top diplomat despite having no government experience surprised senior Exxon officials—including Mr. Tillerson, according to people familiar with the matter.

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