AMI didn't demand any money from Bezos, the world's richest person, only that he call off his investigation and issue a statement saying the coverage wasn't political.
In its January 9 story, the Enquirer reported that Mr. Bezos sent "sleazy text messages and gushing love notes" to Ms. Sanchez, months before Mr. Bezos announced he was splitting up with his wife, MacKenzie.
The National Enquirer published a story last month that included lurid texts between Bezos and former TV anchor Lauren Sanchez.
In a post on blogging platform Medium Thursday, Bezos said Enquirer publisher American Media Inc (AMI), led by David Pecker, approached him with a threat to publish the photos if he did not halt an investigation into the motives behind that leak.
On MSNBC Thursday night, Washington Post reporter Manuel Roig-Franzia said he had "spoken extensively" with the head of Bezos' investigation, security consultant Gavin de Becker.
"In the AMI letters I'm making public, you will see the precise details of their extortionate proposal", Bezos wrote.
A spokesperson and an attorney for AMI did not immediately respond to messages seeking comment.
Bezos, who owns more than $120 billion in Amazon stock, said in his blog: "If in my position I can't stand up to this kind of extortion, how many people can?" AMI has not commented on the matter. Trump has been highly critical of Bezos and the Post's coverage of the White House. Cohen pleaded guilty to several federal crimes and admitted that he had made a deal, on Trump's behalf, in which the Enquirer helped Trump's presidential bid by paying $150,000 USA to former Playboy playmate Karen McDougal to buy and bury her story.
Reporters for the Enquirer followed Bezos and Sanchez "across five states and 40,000 miles" and "tailed them in private jets, swanky limos, helicopter rides, romantic hikes, five-star hotel hideaways, intimate dinner dates and "quality time" in hidden love nests", the tabloid said in a story about the relationship.
The Enquirer was also a major booster of Trump during the presidential campaign.
Hours later, the first Enquirer story on his relationship with Sanchez posted online.
"Rather than capitulate to extortion and blackmail", wrote Mr Bezos, "I've made a decision to publish exactly what they sent me, despite the personal cost and embarrassment they threaten".
Pecker is now cooperating with federal investigators who are looking into the company's involvement with the Trump campaign. President Trump is one of those people, obvious by his many tweets. The company "emphatically rejects any assertion that its reporting was instigated, dictated or influenced in any manner by external forces, political or otherwise", Fine wrote in an email to de Becker's lawyer, Martin Singer, which Bezos shared.
Bezos includes the emails from AMI in his Medium post. "It's ripped right out of the law books".
But the company has admitted in the past that it engaged in what's known as "catch-and-kill" practices to help Trump become president.
"It shows how complicated and unsafe it is to make an agreement with National Enquirer", Levenson said.