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The discovery of drug-target interactions (DTIs) is a pivotal process in pharmaceutical development. Computational approaches are a promising and efficient alternative to tedious and costly wet-lab experiments for predicting novel DTIs from numerous candidates. Recently, with the availability of abundant heterogeneous biological information from diverse data sources, computational methods have been able to leverage multiple drug and target similarities to boost the performance of DTI prediction. Similarity integration is an effective and flexible strategy to extract crucial information across complementary similarity views, providing a compressed input for any similarity-based DTI prediction model. However, existing similarity integration methods filter and fuse similarities from a global perspective, neglecting the utility of similarity views for each drug and target. In this study, we propose a Fine-Grained Selective similarity integration approach, called FGS, which employs a local interaction consistency-based weight matrix to capture and exploit the importance of similarities at a finer granularity in both similarity selection and combination steps. We evaluate FGS on five DTI prediction datasets under various prediction settings. Experimental results show that our method not only outperforms similarity integration competitors with comparable computational costs, but also achieves better prediction performance than state-of-the-art DTI prediction approaches by collaborating with conventional base models. Furthermore, case studies on the analysis of similarity weights and on the verification of novel predictions confirm the practical ability of FGS.