Last year, Cohen pleaded guilty to federal campaign finance violations, telling the court that he made the arrangements for those hush-money deals "in coordination with and at the direction of a candidate for federal office," referring to then-candidate Trump, "for the principal purpose of influencing the election."
Although Trump has insisted "there were no violations of the campaign finance laws by me," Davidson told ABC News the motivation was clearly political.
"You cannot talk about Stormy Daniels and the settlement without talking about 'Access Hollywood,'" Davidson said. "They come hand in hand. It was clear to me that the 'Access Hollywood' tape was the motivating factor in this case resolving."
According to Davidson, Cohen initially missed the deadline to make a payment to Daniels, effectively cancelling their contract, but after The Washington Post published behind-the-scenes video from the reality show in which Trump can be heard bragging about groping women, the deal took on new urgency.
"It defeats the argument that this was done purely for personal reasons," Davidson said. "It was done for political reasons. The natural conclusion is that after the 'Access Hollywood' tape, that something like this could be the straw that broke the camel's back."
Federal prosecutors from the U.S. Attorney's Office for the Southern District of New York implicated Trump in the scheme in a December court filing in which they asserted that Cohen "deceived the voting public by hiding alleged facts that he believed would have had a substantial effect on the election."
Cohen was sentenced to three years in prison for his crimes following his guilty plea, while American Media Inc., the publisher of the National Enquirer, admitted to paying McDougal $150,000 to "suppress the woman's story so as to prevent it from influencing the election" as part of a non-prosecution agreement.
Trump's role in the deals remains opaque. Davidson said he dealt only with Cohen, so he does not know what, if anything, Trump knew about the negotiations.
But according to Davidson, it might have been the deal with Daniels, concluded shortly before Trump's surprise victory over Democratic candidate Hillary Clinton, that opened some of the first fissures in Trump and Cohen's relationship.
The man who once famously said he would "take a bullet" for Trump was left embarrassed and embittered, Davidson said, when despite his years of loyalty, he was not tapped for a position, like White House chief of staff or attorney general, in Trump's administration.
"He confided in me that he was just beside himself, and, in his words, you know, he said, 'Can you f---ing believe, after everything I've done, he's not taking me to Washington?'" Davidson recalled. "He felt that it was a personal embarrassment for him, that he was rejected."
Cohen told a different story when he testified under oath last week before the House Oversight Committee.
"I was extremely proud to be the personal attorney for the President of the United States of America," Cohen said. "I did not want to go to the White House. I was offered jobs."
Appearing to compound the slight, Davidson said, was that Cohen also was upset Trump was apparently slow to reimburse him for the $130,000 payment he later said he made to Daniels on Trump's behalf.
"I think later on that he did confide in me: 'Can you f----ing believe that son of a bitch never paid me back?'" Davidson said. "That he believes he was owed reimbursement and it hadn't yet come."
Lanny Davis, an attorney who represents Cohen, declined to comment for this report.
Davidson, for his part, said he now regrets his involvement in what he said was "supposed to be the easiest deal in the history of man," but he defended his actions in pursuing the agreements on behalf of his clients.
"There's no amount of compensation that would have justified what I've gone through in the last year," Davidson said. "I believe that consenting adults have a right to contract for confidentiality. And that's exactly what happened here. What their motivation was to settle this case was not my concern."
But asked what effect, if any, those deals had on the outcome of the election, he admits he's lost "a little bit" of sleep on the subject.
"I have no idea. I don't know," Davidson said. "We're living in crazy times."