Federal prosecutors in Manhattan have evidence indicating Jeff Bezos’ girlfriend provided text messages to her brother that he then sold to the National Enquirer for its article about the Amazon.com Inc. founder’s affair, according to people familiar with the matter.

The text messages, which were reviewed by The Wall Street Journal, were among the materials turned over to federal prosecutors as part of their investigation into whether American Media Inc., publisher of the National Enquirer, attempted to extort Mr. Bezos, the...