Mr. Davidson’s client list had included the professional athletes Jalen Rose and Manny Pacquiao, as well as gossip-page regulars who placed him in the middle of the sex-tape cases of the “Austin Powers” actor Verne Troyer, the wrestler Hulk Hogan and the onetime Playboy model and MTV host Tila Tequila. He was a natural choice for Ms. Clifford, also known as Stormy Daniels, when she sought to sell her own Trump story.

She was alleging that she had had a consensual sexual relationship with Mr. Trump after they met at a celebrity golf tournament about 10 years earlier (Mr. Trump denies her claims).

Just two months after Ms. McDougal’s story was effectively muted by her contract with American Media, Mr. Davidson set about brokering the silence of the adult film actress. This time, the negotiator on the other end of the transaction was Mr. Cohen.

The actress agreed to a $130,000 settlement in mid-October 2016 in exchange for keeping quiet, according to contracts seen by The Times and people familiar with the matter. To make the payment, Mr. Cohen created a Delaware limited liability company called Essential Consultants, news of which was first reported by The Wall Street Journal last month, and he claimed in a statement first released to The Times last week that the money came from his own pocket.

Ms. Clifford has suggested in recent days that she believes Mr. Cohen has breached that agreement and that she is preparing to speak out. In 2011, she had told her story about Mr. Trump to two gossip publications. One of them, In Touch magazine, did not publish the story after Mr. Cohen warned that he would pursue aggressive legal action, The Associated Press reported last month.

The other outlet, The Dirty, took down a brief story after Mr. Davidson threatened legal action just a day after his client had provided information to the website, according to Nik Richie, The Dirty’s founder, and a letter seen by The Times.

After the deal between Ms. McDougal and A.M.I. was completed, Mr. Davidson regularly exchanged emails, text messages and calls with Mr. Cohen, according to people familiar with the contacts, including last week, when Mr. Davidson publicly bolstered Mr. Cohen’s statement that he had paid Ms. Clifford himself.

Mr. Cohen went on to steer a new client to Mr. Davidson, Chuck LaBella, a former NBC executive who worked closely with Mr. Trump on “The Apprentice” and the “Miss USA” pageant. Mr. LaBella had become the object of an intense Twitter campaign — led by the comedian and ardent Trump critic Tom Arnold — calling upon him to share anything he might know about misbehavior by Mr. Trump. He became a client of Mr. Davidson last fall, according to people familiar with the arrangement.