AMI was apparently seeking a disavowal of statements in a story the Post published Tuesday exploring whether The National Enquirer's coverage of Bezos' affair - which featured intimate text messages between Bezos and Sanchez - was politically motivated.
Bezos, who retained private security consultant Gavin de Becker for his investigation, said an "AMI leader" subsequently told him that David Pecker, the chief executive of AMI, was "apoplectic" about the investigation.
He added: "Rather than capitulate to extortion and blackmail, I've made a decision to publish exactly what they sent me, despite the personal cost and embarrassment they threaten", Bezos wrote.
"Nonetheless, in light of the nature of the allegations published by Mr. Bezos, the Board has convened and determined that it should promptly and thoroughly investigate the claims", the statement continued. It says it'll take "whatever appropriate action is necessary".
Farrow has written extensively about AMI's "catch and kill" practice, in which the publisher buys stories in order to bury them, and how such measures benefited Pecker's pal, Donald Trump, in the 2016 presidential campaign.
In a blog post on the Medium.com website, Bezos revealed what he characterized as "extortion and blackmail" by publishing a series of emails he says were exchanged by his and AMI's attorneys. Soon after, the tabloid carried reports about text messages hinting at the billionaire's affair.
De Becker is now trying to find a way that federal prosecutors in Manhattan - where the non-prosecution agreement was signed - could investigate the text message scandal, the person familiar with the matter said, though it wasn't immediately clear what, if any, crime the prosecutors would be asked to look into.
"So sorry to hear the news about Jeff Bozo being taken down by a competitor whose reporting, I understand, is far more accurate than the reporting in his lobbyist newspaper, the Amazon Washington Post", he tweeted. Bezos' investigators have suggested the Enquirer's coverage of his affair was driven by dirty politics.
"The National Enquirer has been doggedly investigating this story for four months and the extraordinary details and evidence uncovered by our team, and presented to Mr. Bezos' representatives for comment early this week, underscores the kind of investigative reporting that the publication has always been known for", a spokesman says. The federal lawyers agreed not to prosecute AMI in the campaign finance violation case related to Trump and Michael Cohen as a reward for AMI cooperating, Rachel Maddow explains in the video below.
Republican President Donald Trump and Bezos have crossed swords in the past, with Trump criticising both the Washington Post and Amazon, which Bezos owns.
In Mr Bezos' post on Thursday, he also published several e-mails between Mr de Becker's lawyer, Mr Martin Singer, and AMI's lawyer, Mr Jon Fine, and chief content officer, Mr Dylan Howard.
AMI's financial backer, hedge fund manager Anthony Melchiorre, also has rubbed shoulders with the president, including a White House dinner in July 2017.
The law defines extortion as, among other things, obtaining something of value through fear, and AMI's very explicit threat to publish the embarrassing pictures unless Bezos complies, fits that description, according to Elsea.
Jeff Bezos, Amazon founder and CEO and David Pecker, Chairman and CEO of American Media, the company behind the National Enquirer. President Trump is one of those people, obvious by his many tweets. (AMI), the parent company of the National Enquirer. In it, AMI proposed a public acknowledgment from Bezos and de Becker that "they have no knowledge or basis for suggesting that (AMI's) coverage was politically motivated or influenced by political forces".
The Post reported that Michael Sanchez said he was told by multiple people at AMI that the Enquirer set out to do "a takedown to make Trump happy".