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 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 autoencoder's latent space. In the second phase, we propose to employ pairwise 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 autoencoder'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.