Compared with traditional remoting image, there is a large amount of spectral information in the hyperspectral image (HSI), which makes HSI better reflect the actual condition of surface features. However, due to the limitations of imaging conditions, HSI tends to have a lower spatial resolution. In order to overcome this issue, we propose a spectral-spatial attention-based U-Net named SSAU-Net for HSI and multispectral image (MSI) fusion. The SSAU-Net constructs a spectral-spatial attention module by a coordinate-attention (CA) module and an efficient pyramid split attention (ESPA) module, which can enhance the image's spectral information and spatial information. Meanwhile, the proposed network fully extracts the shallow and deep features of the images, and finally generates high-resolution (HR) hyperspectral images. Compared with state-of-the-art HSI-MSI fusion methods, the experimental results verify that the proposed method has a better subjective and objective fusion effect.