Interpreting the decisions of deep learning models has been actively studied since the explosion of deep neural networks. One of the most convincing interpretation approaches is salience-based visual interpretation, such as Grad-CAM, where the generation of attention maps depends merely on categorical labels. Although existing interpretation methods can provide explainable decision clues, they often yield partial correspondence between image and saliency maps due to the limited discriminative information from one-hot labels. This paper develops a Language-Image COnsistency model for explainable image classification, termed LICO, by correlating learnable linguistic prompts with corresponding visual features in a coarse-to-fine manner. Specifically, we first establish a coarse global manifold structure alignment by minimizing the distance between the distributions of image and language features. We then achieve fine-grained saliency maps by applying optimal transport (OT) theory to assign local feature maps with class-specific prompts. Extensive experimental results on eight benchmark datasets demonstrate that the proposed LICO achieves a significant improvement in generating more explainable attention maps in conjunction with existing interpretation methods such as Grad-CAM. Remarkably, LICO improves the classification performance of existing models without introducing any computational overhead during inference.