State-of-the-art speaker verification models are based on deep learning techniques, which heavily depend on the hand-designed neural architectures from experts or engineers. We borrow the idea of neural architecture search (NAS) for the text-independent speaker verification task. As NAS can learn deep network structures automatically, we introduce the NAS conception into the well-known x-vector network. Furthermore, this paper proposes an evolutionary algorithm enhanced neural architecture search method called Auto-Vector to automatically discover promising networks for the speaker verification task. The experimental results demonstrate our NAS-based model outperforms state-of-the-art speaker verification models.