Animal pose estimation is challenging for existing image-based methods because of limited training data and large intra- and inter-species variances. Motivated by the progress of visual-language research, we propose that pre-trained language models (eg, CLIP) can facilitate animal pose estimation by providing rich prior knowledge for describing animal keypoints in text. However, we found that building effective connections between pre-trained language models and visual animal keypoints is non-trivial since the gap between text-based descriptions and keypoint-based visual features about animal pose can be significant. To address this issue, we introduce a novel prompt-based Contrastive learning scheme for connecting Language and AniMal Pose (CLAMP) effectively. The CLAMP attempts to bridge the gap by adapting the text prompts to the animal keypoints during network training. The adaptation is decomposed into spatial-aware and feature-aware processes, and two novel contrastive losses are devised correspondingly. In practice, the CLAMP enables the first cross-modal animal pose estimation paradigm. Experimental results show that our method achieves state-of-the-art performance under the supervised, few-shot, and zero-shot settings, outperforming image-based methods by a large margin. The code is available at https://github.com/xuzhang1199/CLAMP.