Hyperspectral images (HSIs) typically have finer spectral resolution but coarser spatial resolution than multispectral images (MSIs). To obtain HSIs with enhanced spatial resolution, considerable emphasis has been placed on achieving hyperspectral super-resolution (SR) by fusing HSIs with MSIs in the same scene. However, most existing HSI-MSI fusion methods either rely on prior knowledge of degradation models or require sufficient training data, hindering their practicality and interpretability. This letter proposes a deep multistream network (DMSN) for HSI SR. Specifically, we introduce the Spa-DNet and the Spe-UNet modules to encode spatial and spectral transformations across resolutions. Furthermore, we design the Int-Net to achieve spatial and spectral information interaction, enhancing the model’s performance. Finally, the proposed approach enables high spatial and spectral resolution HSIs. Using the newly designed three-stage training strategy, the network parameters can exhibit the clear physical significance of the degradation process, thereby helping to ensure faithful reconstruction of the desired HSIs. Experimental results with real datasets demonstrate that the proposed DMSN performs better than other methods. The codes will be available at https://github.com/yuanchaosu/dmsn-GRSL.