Spectral imaging, with the ability to simultaneously capture the spectral and spatial information of scenes, has obtained researchers' significant interest. Traditional spectral imaging techniques typically suffer from high costs and slow imaging speed. Multispectral filter array-based snapshot imaging is a cutting-edge technology for mitigating these problems. The spectral image demosaicing algorithm plays a key role in reconstructing high-resolution spectral images from the raw measurement. In this article, we first formulate the multispectral demosaicing problem as a compressive spectral imaging problem. Then, a novel deep spectral image prior is introduced as the regularization, which assumes that neural networks can generate the desired spectral image from the raw image in a self-supervised learning manner. Finally, the constructed constrained problem is solved by the alternating direction method of multipliers optimization algorithm. Compared with existing methods, experiments on various scenes demonstrate that the proposed method obtains superior performance.