Circular economy (CE) and sustainable development are not systematically synonyms and some trade-offs might occur between circularity and sustainability. This study addresses the nexus—i.e., the potential concurrence, complementarity, or conflict—between circularity and sustainability indicators to develop and monitor more circular and sustainable systems, using two complementary research approaches: (i) a review and comparison of existing works covering circularity and sustainability performances concurrently (Part I), (ii) new experimentations on more circularity and sustainability to fill the gaps of previous works and bring more concrete insights for practitioners on this matter (Part II). In the present Part I, we provide a state-of-the-art summary of the solutions and challenges related to the measurement of the sustainability performance of CE strategies. The contributions and limitations of life cycle assessment (LCA), leading sustainability, and circularity indicator–based approaches are analyzed, compared, and illustrated. Ten independent studies experimenting and comparing LCA and circularity indicators on different products are analyzed and compared. Depending on the product context and the CE scenarios evaluated, several types of connections emerged: beneficial, conditional, or scenario-dependent trade-offs. To fill the gaps of the existing studies exposed at the end of this paper (Part I), a new set of experimentations is designed and conducted (Part II) to further understand how these different approaches and indicators could be appropriately deployed to develop more circular and sustainable product systems.