“This is a democracy, and this matter should be decided in an open court of law owned by the people,” Mr. Avenatti said.

Mr. Trump and his lawyers have been trying vigorously to avoid further public statements by Ms. Clifford, particularly since she said in February that she believed that Mr. Cohen had violated the agreement and that she, as a result, was no longer bound by it. Mr. Cohen secretly obtained a restraining order late that month to prevent her from speaking.

And last month, Mr. Trump’s legal team filed a motion asking to move the case from state court to federal court, which may have been motivated by a desire to keep the case in arbitration: The Federal Arbitration Act favors arbitration in certain types of disputes, and federal courts have generally applied that law more strictly than state courts.

Also on Monday, America Media Inc. answered a lawsuit from the former Playboy model Karen McDougal, who also claims to have had an affair with Mr. Trump. Ms. McDougal alleged in her lawsuit that the company, which publishes The National Enquirer, misled her when it made a $150,000 deal to squelch her story, buying the exclusive rights to it during the campaign but never publishing anything.

American Media asked the Los Angeles Superior Court to throw Ms. McDougal’s lawsuit out based on an “anti-SLAPP” statute, which many states have enacted to halt frivolous lawsuits devised to “chill the valid exercise of the constitutional right of free speech and petition.” In effect, the company’s lawyers are arguing that Ms. McDougal is challenging its “constitutional and contractual right to exercise its editorial discretion not to publish.”

The company said in a statement on Monday that it hoped Ms. McDougal would stay on as “a valued contributor” and that it still sought “an amicable resolution” with her.

A lawyer for Ms. McDougal, Peter K. Stris, wrote on Twitter, “The tabloid went to great lengths to silence her and others, and they are now attempting to silence her again with the absurd claim that their own free speech was violated.”

Ms. McDougal’s lawsuit alleges that American Media engaged her in the agreement in order to influence the 2016 election. The watchdog group Common Cause has filed complaints with the Justice Department and the Federal Election Commission charging that the $150,000 payment — which also bought the rights to columns, blog posts and cover shoots with Ms. McDougal — was an illegal, in-kind contribution to Mr. Trump’s campaign.

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