Fake news often involves multimedia information such as text and image to mislead readers, proliferating and expanding its influence. Most existing fake news detection methods apply the co-attention mechanism to fuse multimodal features while ignoring the consistency of image and text in co-attention. In this paper, we propose multimodal matching-aware co-attention networks with mutual knowledge distillation for improving fake news detection. Specifically, we design an image-text matching-aware co-attention mechanism which captures the alignment of image and text for better multimodal fusion. The image-text matching representation can be obtained via a vision-language pre-trained model. Additionally, based on the designed image-text matching-aware co-attention mechanism, we propose to build two co-attention networks respectively centered on text and image for mutual knowledge distillation to improve fake news detection. Extensive experiments on three benchmark datasets demonstrate that our proposed model achieves state-of-the-art performance on multimodal fake news detection.