Ms. Clifford went forward with the interview, which her lawyer first hinted about in a tweet featuring a photograph of him, Ms. Clifford and Mr. Cooper. It proceeded despite an arbitrator’s ruling reaffirming an agreement that she reached with Mr. Trump in October 2016 to remain silent about their alleged relationship in exchange for $130,000.

She has said she had a consensual relationship with Mr. Trump that started in 2006 and lasted several months.

In recent weeks, Ms. Clifford’s lawyer, Michael Avenatti, has led a publicity blitz, appearing frequently on cable news shows, since she filed a lawsuit on March 6 seeking to break her 2016 agreement. She asserts that the nondisclosure agreement was void because Mr. Trump did not personally sign it.

That lawsuit came after Michael D. Cohen, the president’s longtime personal lawyer who said he paid her $130,000 legally pressured her to stay silent. Days before Ms. Clifford sued, Mr. Cohen secretly obtained a temporary restraining through an arbitrator in California to prevent her from speaking about her alleged affair.

Mr. Avenatti wrote in a letter to Mr. Cohen on Monday that Ms. Clifford would be willing to pay back the $130,000 to end the deal to stay quiet. The offer had a deadline of noon Tuesday for Mr. Cohen to answer, which came and went.