The alleged hack deepens the intrigue around how the Enquirer - which had close links with U.S. President Donald Trump at the time of the alleged extortion - obtained messages exchanged between Bezos and Lauren Sanchez, an ex-TV anchor who the tabloid said he was dating.
Last year Bezos' security chief said the Saudi government was the source of the messages. A month before, Bezos had accused the Enquirer's owner of trying to blackmail him with the threat of publishing "intimate photos" he allegedly sent to Sanchez.
All this took place as Trump - who has closely allied the United States with bin Salman - was clashing with Bezos and the Washington Post in public.
"So sorry to hear the news about Jeff Bozo being taken down by a competitor," Trump gloated on Twitter as news of the clash between Bezos and the Enquirer went public. Trump also took the opportunity to take a shot at the Post: "Hopefully the paper will soon be placed in better & more responsible hands!"
The Saudi government has denied having anything to do with the National Enquirer's reporting.
The Guardian newspaper first reported the crown prince's alleged involvement in the phone hacking targeting Bezos.