Anomalous sound detection for machine condition monitoring has great potential in the development of Industry 4.0. However, these anomalous sounds of machines are usually unavailable in normal conditions. Therefore, the models employed have to learn acoustic representations with normal sounds for training, and detect anomalous sounds while testing. In this article, we propose a self-supervised dual-path Transformer (SSDPT) network to detect anomalous sounds in machine monitoring. The SSDPT network splits the acoustic features into segments and employs several DPT blocks for time and frequency modeling. DPT blocks use attention modules to alternately model the interactive information about the frequency and temporal components of the segmented acoustic features. To address the problem of lack of anomalous sound, we adopt a self-supervised learning approach to train the network with normal sound. Specifically, this approach randomly masks and reconstructs the acoustic features, and jointly classifies machine identity information to improve the performance of anomalous sound detection. We evaluated our method on the DCASE2021 task2 dataset. The experimental results show that the SSDPT network achieves a significant increase in the harmonic mean AUC score, in comparison to present state-of-the-art methods of anomalous sound detection.