Dense hand contact estimation requires both high-level semantic understanding and fine-grained geometric reasoning of human interaction to accurately localize contact regions. Recently, multi-modal large language models (MLLMs) have demonstrated strong capabilities in understanding visual semantics, enabled by vision-language priors learned from large-scale data. However, leveraging MLLMs for dense hand contact estimation remains underexplored. There are two major challenges in applying MLLMs to dense hand contact estimation. First, encoding explicit 3D hand geometry is difficult, as MLLMs primarily operate on vision and language modalities. Second, capturing fine-grained vertex-level contact remains challenging, as MLLMs tend to focus on high-level semantics rather than detailed geometric reasoning. To address these challenges, we propose ContactPrompt, a training-free and zero-shot approach for dense hand contact estimation using MLLMs. To effectively encode 3D hand geometry, we introduce a detailed hand-part segmentation and a part-wise vertex-grid representation that provides structured, localized geometric information. To enable accurate and efficient dense contact prediction, we develop a multi-stage structured contact reasoning with part conditioning, progressively bridging global semantics and fine-grained geometry. Therefore, our method effectively leverages the reasoning capabilities of MLLMs while enabling precise dense hand contact estimation. Surprisingly, the proposed approach outperforms previous supervised methods trained on large-scale dense contact datasets without requiring any training. The codes will be released.