Accurate sclera segmentation is critical for successful sclera recognition. However, studies on sclera segmentation algorithms are still limited in the literature. In this paper, we propose a novel sclera segmentation method based on the improved U-Net model, named as ScleraSegNet. We perform in-depth analysis regarding the structure of U-Net model, and propose to embed an attention module into the central bottleneck part between the contracting path and the expansive path of U-Net to strengthen the ability of learning discriminative representations. We compare different attention modules and find that channel-wise attention is the most effective in improving the performance of the segmentation network. Besides, we evaluate the effectiveness of data augmentation process in improving the generalization ability of the segmentation network. Experiment results show that the best performing configuration of the proposed method achieves state-of-the-art performance with F-measure values of 91.43%, 89.54% on UBIRIS.v2 and MICHE, respectively.