Although Clifford has been making public appearances around the country, she's remained silent about the alleged 2006 affair with the president after receiving the hush money from Cohen just before the 2016 US presidential election.
The 38-year-old Daniels, whose real name is Stephanie Clifford and who turns 39 on March 17, allegedly had a long-running sexual relationship with Trump in 2006 and 2007 and received a $130,000 "hush money" payment from Trump's personal lawyer and self-described "fix it guy", Michael Cohen, just weeks before the 2016 presidential election.
Cohen has said he made the payment out his pocket and neither the Trump Organization nor the Trump campaign was a party to the transaction. The White House declined to comment.
Last week, White House press secretary Sarah Huckabee said "arbitration was won in the President's favor", language that suggested Trump was involved.
If they accept, Ms Clifford could "speak openly and freely about her prior relationship with the president and the attempts to silence her and use and publish text messages, photos and videos relating to the president that she may have in her possession, all without fear of retribution or legal liability", the letter said.
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By sending the money to Trump, it would involve the President - who has never acknowledged the agreement or the affair - in the case.
"As we have always said, this is about a search for the truth and the ability of Ms Clifford to tell the American people what really happened so they can make their own determination, " Mr Avenatti said in a statement to the New York Times, who first reported the news.
Emails to Avenatti and Cohen seeking comment were not immediately returned.
Trump can hope that the prospect of a libel suit will scare CBS out of airing the interview, or he can hope that further reporting will cause 60 Minutes to conclude that Daniels is not credible and thus abandon the interview for journalistic reasons.
The New York Times reported last week that the president's lawyer had secretly obtained a temporary restraining order to prevent Clifford from speaking out about her alleged affair with Trump. (PDF)Los Angeles-based lawyer Michael Avenatti told CNN's Michael Smerconish that no California court would enforce the $1 million penalty for "every episode that she goes out and speaks in violation of the confidentiality".