Deep clustering incorporates embedding into clustering to find a lower dimensional space appropriate for clustering. In this paper, we propose a novel deep clustering framework with self-supervision using pairwise data similarities (DCSS). The proposed method consists of two successive phases. In the first phase, we propose to form hypersphere-like groups of similar data points, i.e. one hypersphere per cluster, employing an autoencoder that is trained using cluster-specific losses. The hyper-spheres are formed in the autoencoderâ\euro™s latent space. In the second phase, we propose to employ pairwise data similarities to create a K-dimensional space that is capable of accommodating more complex cluster distributions, hence providing more accurate clustering performance. K is the number of clusters. The autoencoderâ\euro™s latent space obtained in the first phase is used as the input of the second phase. The effectiveness of both phases is demonstrated on seven benchmark datasets by conducting a rigorous set of experiments. The DCSS code is available: https://github.com/Armanfard-Lab/DCSS