The precise segmentation of medical images is crucial for clinical applications like computer-aided diagnosis and computer-aided surgery. However, dealing with the blurred edges of the region of interest poses challenges not only in terms of data annotation limitations but also for segmentation tasks. Edge-Net, a self-supervised model with edge attention, was proposed to address these issues. Through an innovative edge-aware attention mechanism, the model automatically learns key information of the target boundary. This approach has achieved the best results on two publicly medical image datasets. On the Abdomen and CHAOS datasets, the Dice coefficients are 78.03% and 77%, the HD95 are 21.17 mm and 30.26 mm, and the ASSD are 6.43 mm and 9.44 mm, respectively. It greatly alleviates the problem that self-supervised learning is easily affected by wrong label information, and provides an effective solution for self-supervised learning of medical images.