Yes, it's true as per Ramcharitmanas: Ayodhya Kanda: 11-12Ramcharitmanas: Ayodhya Kanda: 11-12
Here are few relevant chopais, Dohas:
सारद बोलि बिनय सुर करहीं। बारहिं बार पाय लै परहीं।।
Invoking Sharda (Maa Saraswati) the gods supplicated her and laying hold of her feet fell at them again and again.
बिपति हमारि बिलोकि बड़ि मातु करिअ सोइ आजु।
रामु जाहिं बन राजु तजि होइ सकल सुरकाजु।।11।।
Perceiving our grave calamity, O Mother, manipulate things in such a way today that Shri Rama may retire into the forest, relinquishing His throne, and the object of us immortals may be wholly accomplished.
नामु मंथरा मंदमति चेरी कैकेइ केरि।
अजस पेटारी ताहि करि गई गिरागिरा मति फेरि।।12।।
Kaikeyi had a dull-witted servant-maid, Manthara by name; having perverted her reason and making her a receptacle of ill-repute, the goddess of speechthe goddess of speech (Maa Saraswati) returned to her abode.