WASHINGTON—Hillary Clinton is proposing an expansive program aimed at enabling students to attend public colleges and universities without taking on loans for tuition, her attempt to address a source of anxiety for American families while advancing one of the left’s most sweeping new ideas.

The plan—dubbed the “New College Compact” and estimated to cost $350 billion over 10 years—would fundamentally reshape the federal government’s role in higher education by offering new federal money, but with strings attached.

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