Davidson also filed a counterclaim against Daniels and her current lawyer Michael Avenatti, accusing them of defaming him.
Davidson's allegations came in response to a lawsuit that Daniels filed Wednesday against him and Cohen in California federal court.
In that case, Daniels accuses Davidson of colluding with Cohen this year, without her knowledge, in an effort to get her to publicly falsely state that she did not have an affair with Trump in 2006.
That alleged collusion came more than a year after a shell company set up by Cohen paid Daniels $130,000 in exchange for what she has said was her silence about the tryst with Trump.
The payment, which Davidson negotiated with Cohen on Daniels' behalf, was made in October 2016, just weeks before the election that sent Trump to the White House.
Trump later repaid Cohen for the payment to Daniels — but spokesmen for the president said the affair never happened.
In a court filing Thursday, Davidson alleged that Cohen, whose law office is in New York City, "surreptitiously and intentionally recorded several telephone calls" with Davidson during their negotiations over the hush deal.
Davidson said the alleged recordings, made while he was in California, violated that state's Invasion of Privacy Act, which requires the consent of all parties involved to record private conversations.
Multiple media outlets have reported that the FBI seized recordings in raids on Cohen's home, office and hotel room in April as part of an ongoing criminal investigation. Cohen has not been charged with any crime.
Davidson wants Cohen to pay at least $75,000 for the costs of the lawsuit.
Brent Blakely, a lawyer for Cohen, did not immediately respond to a request for comment on Davidson's allegation.
Davidson's related defamation claim against Daniels and Avenatti accuses the porn star's current lawyer of conducting a "crusade to falsely vilify Davidson while using Clifford's Attorney Client Privilege with Davidson as both a sword and a shield."
Noting that Avenatti has made more than 175 appearances on television in recent months, Davidson said Avenatti has made "countless reckless and false statements" against him.
But Davidson only cited one specific example in his court filing: a tweet by Avenatti on Wednesday saying Davidson "should have been charged after his arrest for extortion."
That tweet is linked to a Daily Beast story from April, which reported that Davidson was arrested in 2012 after soliciting money from wrestler Hulk Hogan in exchange for all copies of a surreptitiously recorded sex tape of Hogan.
However, Davidson was not charged with a crime in that case, according to the Daily Beast story, which cited court documents.
Davidson in his court filing Thursday said that as a result of statements by Daniels and Avenatti he has "suffered injury to his personal, business and professional reputation including suffering embarrassment, humiliation, severe emotional distress, shunning, anguish, fear, loss of employment, and employability, and significant economic loss and future earnings, all to his economic, emotional and general damage."
Avenatti fired back at Davidson in a statement to CNBC.
"There is no question that at the end of this, Keith Davidson will be disbarred from the practice of law," Avenatti said.
"He is a proven liar and his conduct is abhorrent. He is an embarrassment to the profession and seems to have forgotten what the attorney-client privilege is all about. We look forward to having his frivolous claims thrown out of court."
In her suit filed Wednesday alleging collusion by her ex-lawyer and the president's attorney, Daniels cites text messages between Cohen and Davidson, showing an attempt by Cohen to get her to appear in January on the Fox News program "Hannity."
Avenatti says the attempted booking was meant for her to publicly deny the alleged affair with Trump. At the time, In Touch magazine was set to publish for the first time an interview with Daniels from years before in which she described how she came to have sex with Trump.
The "Hannity" appearance never happened. In a text cited in Daniels' suit, Cohen told Davidson: "Keith, The wise men all believe the story is dying and don't think it's smart for her to do any interviews. Let her do her thing but no interviews at all with anyone."
Avenatti alleges the "wise men" included Trump. But the suit does not offer any proof for that claim.
Daniels's suit alleges that Davidson has been withholding documents from her to which she is entitled.
Davidson's spokesman Dave Wedge, said, "It is an absolute blatant lie that Attorney Davidson has not given Mr. Avenatti the entire client file."
He added: "It's preposterous and completely baseless for Mr. Avenatti to suggest that any materials have been destroyed. Mr. Avenatti has already defamed Attorney Davidson and is continuing to use Twitter and the media to spread lies and misinformation as he continues his desperate 'publicity tour.'"