The New York Department of Financial Services has sent letters to Facebook Inc. and mobile app developers in response to Gov. Andrew Cuomo’s request that it probe potential uses of third-party app data without consent, according to a person familiar with the investigation.

Cuomo (D) called on New York’s departments of State and Financial Services to investigate following a Wall Street Journal report that Facebook accesses sensitive personal information sent to the company by apps without users’ consent.

Facebook is...