President Trump and a company affiliated with him filed court papers on Monday seeking to force the pornographic film actress Stephanie Clifford to raise her disputes through private arbitration, not lawsuits.

Ms. Clifford, whose stage name is Stormy Daniels, was paid $130,000 to keep quiet about her claims of an affair with Mr. Trump. She sued last month to get out of the nondisclosure agreement she signed in October 2016, alleging that it was void because Mr. Trump had never signed it. That raised the prospect of further embarrassing revelations for a president who has already been rocked by Ms. Clifford’s public statements.

In a motion filed on Monday, Essential Consultants — a shell company established by Mr. Trump’s lawyer Michael D. Cohen to pay Ms. Clifford — asked the United States District Court for the Central District of California to compel Ms. Clifford “to arbitrate any and all disputes arising” from the confidentiality agreement she signed. In a separate document filed on the same day, Mr. Trump formally joined the motion.