WASHINGTON—Securities regulators have inquired with Tesla Inc. about Chief Executive Elon Musk’s surprise announcement that he may take the company private and whether his claim was factual, people familiar with the matter said.

The Securities and Exchange Commission has asked whether Mr. Musk’s unusual announcement on Tuesday was factual, the people said. The regulator also asked Tesla about why the disclosure was made on Twitter rather than in a regulatory filing, and whether the company believes the announcement complies...