The promising applications of large language models are often constrained by the limited GPU memory capacity available on edge devices. Mixture-of-Experts (MoE) models help mitigate this issue by activating only a subset of the model's parameters during computation, allowing the unused parameters to be offloaded to host memory and reducing overall GPU memory demand. However, existing cache-based offloading solutions handle cache misses reactively and significantly impact system performance. In this paper, we propose ProMoE, a novel proactive caching system that leverages intermediate model results to predict subsequent parameter usage. By proactively fetching experts in advance, ProMoE removes the loading time from the critical path and diminishes the performance overhead of offloading. Our evaluations demonstrate that ProMoE achieves an average speedup of 2.13x and 2.84x in the prefill and decode stages respectively, compared to existing offloading solutions.