Multi-view subspace clustering has attracted widespread attention for its superior clustering effectiveness. However, most of the current methods do not fully exploit the intra-view (sample-to-sample) and inter-view (view-to-view) information among multi-view data. Focusing on this problem, we propose a method called enhanced intra-inter view correlation learning for multi-view subspace clustering (EIVCL). EIVCL simultaneously considers the intra-view and inter-view correlations and introduces two constraint terms for each aspect to capture the data structure information. Specifically, in the intra-view space, we apply the tensor-singular value decomposition (t-SVD)-based tensor nuclear norm (TNN) on the tensor formed by stacking self-representative coefficient matrices for obtaining the high-order correlation information between samples in the specific view. Moreover, we introduce a hypergraph-induced Laplace regularization term to preserve the local geometric structure within views. In the inter-view space, we impose the t-SVD-based TNN on the rotated tensor to obtain the multiple views correlation. Furthermore, we utilize the kernel dependence metric, namely Hilbert–Schmidt independence criterion, to capture the high-order non-linear relationships between views. In addition, all above strategies are integrated into a unified clustering framework, which is solved by our proposed optimization algorithm based on the alternating direction method of multipliers. Extensive experiments on six benchmark datasets demonstrate that EIVCL outperforms several state-of-the-art multi-view algorithms.