"The claim we held any reporting regarding Stormy Daniels is false and outrageous," said Steve Severinghaus, the senior communications director of Dow Jones, the company that publishes the Journal. Dow Jones is owned by News Corp, the global media conglomerate headed by Rupert Murdoch.
"In fact, the Journal broke the news of the $130k payout to her, arranged by Michael Cohen," Severinghaus added.
Additionally, the Journal did publish a story before the election detailing a $150,000 payment made to former Playboy model Karen McDougal, who was represented by Davidson at the time she signed a deal barring her from discussing her own alleged affair with Trump.
McDougal, who later sued American Media to be released from the deal, alleged in a court filing that Davidson was in contact with Cohen during that negotiation process.
That story came on Nov. 4, one day after Dow Jones responded to Davidson.
As evidence for his accusation, Avenatti linked an attachment to his tweet showing email correspondence from before the 2016 election that appeared to be between Davidson and a Wall Street Journal reporter, as well an email purportedly from Dow Jones to Davidson.
Avenatti said the emails show the Journal failed to report the story before the election. "The documents are clear as day as to what happened. The WSJ had the story and sat on it until 14 months after the election, when they finally broke it."
While the Journal's story on Cohen's payment to Daniels was not published until 2018, there is no evidence in the emails Avenatti published that the newspaper "sat on" the report.
Avenatti told CNBC the documents he published "come directly from Mr. Davidson's files. They are accurate and complete." Asked for more details on how he obtained the emails, Avenatti said he "demanded" them and they were provided through "the rules of professional conduct."
The attachment shows that a Journal reporter reached out to Davidson on Oct. 21, 2016, asking to speak with him for a story. On Nov. 2 — less than a week before the 2016 election — Davidson responded by demanding that the newspaper "refrain from publishing, distributing or disseminating any factually untrue and unsubstantiated information regarding me or my firm" or face legal action.
The attachment also shows that the following day, Dow Jones' Associate General Counsel Craig Linder told Davidson that the reporters would continue to investigate despite the legal threat.
Linder did not respond to CNBC's request for comment. Dave Wedge, a spokesman for Davidson, said Davidson is "unable at this time to respond point-by-point to each one of the numerous false and misleading accusations made by him over the last several months."
Wedge added: "Again today, Mr. Avenatti used Twitter to launch a defamatory charge against Attorney Davidson, who has been and shall continue to be a zealous advocate for the best interests of his clients. Attorney Davidson looks forward to responding to these scurrilous accusations in an appropriate manner, which does not include Twitter."
Avenatti is currently vying to be able to represent Daniels in court proceedings regarding a raft of materials seized from Cohen's properties in April by federal agents. Some of the seized materials may be related to Avenatti's client, whose real name is Stephanie Clifford. The judge in those proceedings could make a decision on Avenatti's request as early as Wednesday.
A lawyer for Cohen did not immediately respond to CNBC's request for comment on Avenatti's disclosure.
Avenatti's attack on the Journal comes a day after the newspaper reported that he has "slowed prosecutors' efforts to discuss the nondisclosure agreement with Ms. Clifford's former lawyer," citing people familiar with the matter.
In a tweet earlier on Tuesday, Avenatti shot down the accusation as being "completely false and without basis."
Avenatti said the disclosure did not come in response to the Journal's earlier story, "but it certainly undercuts the credibility of the WSJ and their reporting relating to our case and Mr. Davidson," he added.