WASHINGTON—The Trump administration is moving to require colleges and universities to publish more detailed data on the finances of their graduates, part of a broader effort to make higher education more market driven and focused on consumer choice.

The new rules, set to be unveiled by the Education Department in coming weeks, will build out an Obama-era initiative called the College Scorecard, but will supplant an Obama-era plan known as the “gainful employment” rule. That provision focused government oversight on for-profit...