Bezos, the world's richest person, owns The Washington Post, a frequent target of Trump as he assails the US media as "enemy of the people" and a source of fake news.
The Post was also one of the leading news outlets in coverage of the murder last year of Saudi journalist Jamal Khashoggi, a Post columnist.
Trump has refused to criticize Saudi Crown Prince Mohammed bin Salman, who has been widely accused of ordering the murder.
Last month Trump took aim at Bezos -- referring to him as 'Bozo' -- in what appeared to be an allusion to National Enquirer reporting of the billionaire's relationship with former news anchor and entertainment reporter Lauren Sanchez.
"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.
"Hopefully the paper will soon be placed in better & more responsible hands!"
The National Enquirer accessed private text messages and last month reported that Bezos had an extramarital affair with Sanchez -- a leak that led to his divorce.
On Thursday, Bezos said the tabloid's parent company had threatened to publish intimate photographs he sent to his mistress.
In a post on the online platform Medium, Bezos said Enquirer publisher American Media Inc (AMI), led by David Pecker, a friend of Trump, had threatened to publish the photos if he did not halt an investigation into the motives behind the leak.
The Enquirer had demanded that he and security consultant Gavin de Becker, who is leading the probe, publically state they had "no knowledge or basis for suggesting that AMI's coverage was politically motivated or influenced by political forces", Bezos added.
De Becker mentioned in a recent Daily Beast interview that "strong leads point to political motives" -- and that he was interested in Lauren Sanchez's brother Michael, a vocal supporter of Trump with links to his inner circle, as a possible perpetrator.
In his Medium post, Bezos pointed to AMI and Pecker's previous cooperation with Trump -- including payments made to suppress negative stories, currently under investigation by federal prosecutors. One involves a former Playboy model who said she had an affair with Trump.
The hush-money payment, and a similar one to another woman, was made on the eve of Trump's 2016 presidential election victory.
Trump is suspected of campaign finance violations because of the disbursements on grounds they were made to affect the outcome of the vote and should therefore have been reported to government campaign monitors.
In a deal announced with federal prosecutors late last year, AMI said it would cooperate with those prosecutors. It admitted that before the election it had paid $150,000 to the ex-model, Karen McDougal, to silence her allegations of an affair with Trump. Prosecutors said they would not charge AMI.
Bezos, his newspaper and Amazon are all regular targets of Trump on Twitter.
Bezos, in his post, also raised what he called AMI's links to Saudi Arabia.
"My ownership of the Washington Post is a complexifier for me. It's unavoidable that certain powerful people who experience Washington Post news coverage will wrongly conclude I am their enemy," Bezos wrote in the blog post.
"President Trump is one of those people, obvious by his many tweets. Also, The Post's essential and unrelenting coverage of the murder of its columnist Jamal Khashoggi is undoubtedly unpopular in certain circles."
He added: "Rather than capitulate to extortion and blackmail, I've decided to publish exactly what they sent me, despite the personal cost and embarrassment they threaten," Bezos wrote. His post, entitled "No thank you, Mr. Pecker", included copies of emails from AMI.
"Of course I don't want personal photos published, but I also won't participate in their well-known practise of blackmail, political favors, political attacks, and corruption. I prefer to stand up, roll this log over, and see what crawls out," he added.