The lawsuit, in Los Angeles Superior Court, comes two weeks after Ms Daniels sued Mr Trump to invalidate her own confidentiality agreement.
The report says Ms Daniels, whose real name is Stephanie Clifford, was being “truthful” in saying she and Mr Trump had unprotected sex.
Ms McDougal is scheduled to give an interview on Thursday to CNN‘s Anderson Cooper, while 60 Minutes is scheduled to air an interview with Ms Daniels on Sunday.
She first spoke publicly in October 2016 at a news conference with other women accusing Mr Trump of misconduct.