PARIS—As pharmaceutical giants edge closer to a potential vaccine for the new coronavirus, governments demanding access to any supplies are running up against a hard reality: the bill.

The tension was cast into relief last Thursday when Sanofi SA chairman Serge Weinberg took a call from French Prime Minister Édouard Philippe, according to a person familiar with the matter. Mr. Philippe wanted to know why the chief executive of the company—one of France’s corporate crown jewels—had told an interviewer the U.S. would be first…

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