High resolution (HR) magnetic resonance images (MRI) or computed tomography (CT) images can provide clearer anatomical details of human body, which facilitates early diagnosis of the diseases. However, due to the imaging system, imaging environment and human factors, it is difficult to obtain clear high-resolution images. In this paper, we proposed a novel medical image super resolution (SR) reconstruction method via multi-scale information distillation (MSID) network in the non-subsampled shearlet transform (NSST) domain, namely NSST-MSID network. We first proposed a MSID network that mainly consisted of a series of stacked MSID blocks to fully exploit features from images and effectively restore the low resolution (LR) images to HR images. In addition, most previous methods predict the HR images in the spatial domain, producing over-smoothed outputs while losing texture details. Thus, we viewed the medical image SR task as the prediction of NSST coefficients, which make further MSID network keep richer structure details than that in spatial domain. Finally, the experimental results on our constructed medical image datasets demonstrated that the proposed method was capable of obtaining better peak signal to noise ratio (PSNR), structural similarity (SSIM) and root mean square error (RMSE) values and keeping global topological structure and local texture detail better than other outstanding methods, which achieves good medical image reconstruction effect.