Contact-rich manipulation requires policies to generate precise actions by reasoning over contact forces, robot configurations, and interaction histories beyond visual observations. Existing methods passively condition on force feedback rather than actively predicting future contact dynamics, limiting their ability to generate high-precision actions. To address this problem, we introduce Predictive Sensorimotor Representation (PSR) learning, a framework that learns a hierarchy of predictive representations from multimodal sensorimotor signals and integrates them into the action stream of a visuomotor policy. Specifically, during a pretraining stage, a multimodal Transformer is trained to learn a hierarchy of predictive representations by jointly forecasting future interaction dynamics. The learned hierarchy subsequently augments the action stream, enabling the resulting policy to exploit contact-relevant cues at multiple depths. We further instantiate PSR within a Vision-Language-Action (VLA) model, resulting in PSR-VLA, and evaluate it on six real-world contact-rich manipulation tasks. Experimental results show that PSR-VLA achieves 91.7% overall success, improving over $π_{0.5}$, ForceVLA-$π_{0.5}$, and ForceVLA2-$π_{0.5}$ by 30.0, 22.5, and 19.2 percentage points, respectively. These results demonstrate the effectiveness of the proposed PSR for force-aware, contact-rich manipulation. Videos of the tasks and stability tests are available at https://psr-vla.pages.dev/.

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