The lawyer for adult film actress Stormy Daniels claimed on Friday that he was representing three more women paid hush money to conceal affairs with Donald Trump.
"Time for Michael Cohen and Donald Trump to come 100 percent clean with the American people," Michael Avenatti tweeted in a reference to Trump's former personal attorney.
"All the documents, all the tapes. NOW," Mr Avenatti added. "No more lies or lip service."
The attorney, speaking to reporters late on Thursday in Los Angeles, said the three women "were paid hush money prior to the 2016 election."
Ms Daniels, whose real name is Stephanie Clifford, is suing Mr Trump to invalidate a non-disclosure agreement that prevents her from discussing their alleged intimate relationship that began in 2006.
Mr Cohen, Trump's attorney, paid Daniels $130,000 just days before the presidential election in exchange for her silence.
A recording surfaced this week of Mr Trump and Mr Cohen discussing how to buy the rights to the story of another woman, a Playboy model who claims she had an affair with Mr Trump in 2006 as well.
Mr Cohen, who no longer works for Mr Trump, is under investigation in New York for his business dealings and reportedly whether hush payments violated campaign finance laws.
The tape involves Karen McDougal, a former Playboy model who says she had a months long fling with Mr Trump after they met 12 years ago.
On the tape, Mr Trump and Mr Cohen talk about buying the rights to McDougal's story, which she had sold to the National Enquirer for $150,000. The tabloid never published the story.
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Mr Cohen, meanwhile, is asking a federal judge in Los Angeles to consider a gag order against Mr Avenatti, and to grant Mr Cohen another 90-day delay in proceedings in Daniels's lawsuit.
US District Judge S James Otero heard arguments in the case but made no immediate ruling, and gave no indication of when he might announce his decision.
"We are confident, based on his statements, that the efforts by the president of the United States and Michael Cohen to silence me and prevent me from presenting the truth and evidence to the American people, that those efforts are going to fail as they should," Mr Avenatti said after the hearing.
Mr Cohen's lawyer, Brent H Blakely, argues that a restraining order is necessary because Mr Avenatti has appeared more than 170 times on television and posted over 500 tweets smearing Mr Cohen in order to litigate his case "in the court of public opinion," according to documents filed in the case.
Judge Otero ordered a three-month stay of the case in April in light of a federal criminal investigation of Mr Cohen. The lawsuit also accuses Mr Cohen of defamation over comments he made about Ms Daniels's truthfulness.
Ms Daniels filed a separate defamation suit against Mr Trump in New York, alleging the president libelled her by saying she lied about a man she claims threatened her to force her to keep quiet about the alleged affair with Mr Trump.
"I think there's a reason why the press and the media continue to invite us to come on television and talk about this case," Mr Avenatti added.
"And that is because there's a significant interest, and there should be, because this is about a cover-up at the highest level of our government, which is always an incredibly important thing in the United States."