The Securities and Exchange Commission is investigating the real-estate company run by the family of President Donald Trump’s senior adviser and son-in-law Jared Kushner for its use of a federal investment-for-visa program known as EB-5, according to people familiar with the matter.

In May 2017, Kushner Cos. received a subpoena from the SEC requesting information about its use of the program, one of these people said. As The Wall Street Journal reported last year, that month the company received a separate subpoena from New...