Research on speech-to-speech translation (S2ST) has progressed rapidly in recent years. Many end-to-end systems have been proposed and show advantages over conventional cascade systems, which are often composed of recognition, translation and synthesis sub-systems. However, most of the end-to-end systems still rely on intermediate textual supervision during training, which makes it infeasible to work for languages without written forms. In this work, we propose a novel model, Textless Translatotron, which is based on Translatotron 2, for training an end-to-end direct S2ST model without any textual supervision. Instead of jointly training with an auxiliary task predicting target phonemes as in Translatotron 2, the proposed model uses an auxiliary task predicting discrete speech representations which are obtained from learned or random speech quantizers. When a speech encoder pre-trained with unsupervised speech data is used for both models, the proposed model obtains translation quality nearly on-par with Translatotron 2 on the multilingual CVSS-C corpus as well as the bilingual Fisher Spanish-English corpus. On the latter, it outperforms the prior state-of-the-art textless model by +18.5 BLEU.