CHICAGO—More than a year before Moody’s Investors Service downgraded Chicago’s bonds to junk status, one of its senior analysts asked top city officials to explain why the third-largest U.S. city was healthier than a troubled island commonwealth flirting with insolvency, according to people familiar with the conversation.

“Help me understand why Chicago is different than Puerto Rico?” said the Moody’s analyst, Rachel Cortez, during a February 2014 meeting that Mayor Rahm Emanuel attended, two of these people said. A spokesman...