Richard Goudis resigned last month as chief executive of Herbalife Nutrition Ltd. after a recording of comments he made years ago about bypassing internal accounting policies recently ended up in the hands of federal investigators, according to people familiar with the matter.

When Mr. Goudis was Herbalife’s chief financial officer about a decade ago, he told a colleague working in Hong Kong or mainland China to ignore the company’s expense account limit on entertainment spending, the people said.

The...