Recent studies have shown that deep networks can achieve promising results for image denoising. However, how to simultaneously incorporate the valuable achievements of traditional methods into the network design and improve network interpretability is still an open problem. To solve this problem, we propose a novel model-based denoising method to inform the design of our denoising network. First, by introducing a non-linear filtering operator, a reliability matrix, and a high-dimensional feature transformation function into the traditional consistency prior, we propose a novel adaptive consistency prior (ACP). Second, by incorporating the ACP term into the maximum a posteriori framework, a model-based denoising method is proposed. This method is further used to inform the network design, leading to a novel end-to-end trainable and interpretable deep denoising network, called DeamNet. Note that the unfolding process leads to a promising module called dual element-wise attention mechanism (DEAM) module. To the best of our knowledge, both our ACP constraint and DEAM module have not been reported in the previous literature. Extensive experiments verify the superiority of DeamNet on both synthetic and real noisy image datasets.