WASHINGTON—Treasury Secretary Steven Mnuchin indicated that the U.S. government would take stakes in airline carriers in exchange for billions in direct grants to the companies, part of a $2 trillion economic rescue package, according to people familiar with the matter.

He detailed his plans during last-minute negotiations when the aid to airlines emerged as a major sticking point. Republicans had rejected providing cash grants to airlines, and an earlier version of the legislation would have provided $50 billion in loans…

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