Contact-rich tasks present significant challenges for robotic manipulation policies due to the complex dynamics of contact and the need for precise control. Vision-based policies often struggle with the skill required for such tasks, as they typically lack critical contact feedback modalities like force/torque information. To address this issue, we propose FoAR, a force-aware reactive policy that combines high-frequency force/torque sensing with visual inputs to enhance the performance in contact-rich manipulation. Built upon the RISE policy, FoAR incorporates a multimodal feature fusion mechanism guided by a future contact predictor, enabling dynamic adjustment of force/torque data usage between non-contact and contact phases. Its reactive control strategy also allows FoAR to accomplish contact-rich tasks accurately through simple position control. Experimental results demonstrate that FoAR significantly outperforms all baselines across various challenging contact-rich tasks while maintaining robust performance under unexpected dynamic disturbances. Project website: https://tonyfang.net/FoAR/