Contrastive-based clustering models usually rely on a large number of negative pairs to capture uniform representations, which requires a large batch size and high computational complexity. In contrast, some self-supervised methods perform non-contrastive learning to capture discriminative representations only with positive pairs, but suffer from the collapse of clustering. To solve these issues, a novel end-to-end self-supervised clustering model is proposed in this paper. The basic self-supervised learning network is first modified, followed by the incorporation of a Softmax layer to obtain cluster assignments as data representation. Then, adversarial learning on the cluster assignments is integrated into the methods to further enhance discrimination across different clusters and mitigate the collapse between clusters. To further encourage clustering-oriented guidance, a new cluster-level discrimination is assembled to promote clustering performance by measuring the self-correlation between the learned cluster assignments. Experimental results on real-world datasets exhibit better performance of the proposed model compared with the existing deep clustering methods.