The Hubbard model is the fundamental model for interacting many-body systems in condensed matter physics, providing valuable insights into phenomena such as high-temperature superconductors like cuprates and nickelates. In our work, we apply numerical techniques—dynamical mean-field theory (DMFT), and the dynamical vertex approximation (DΓA)—to investigate the model with a focus on entanglement. We pursue two main approaches. First, we analyze bipartite entanglement by calculating the two-site reduced density matrix from two- and four-point Green’s functions, allowing us to evaluate entanglement measures such as the negativity and mutual information. Second, we propose an adaptation to the quantum variance as an experimentally accessible bound to the quantum Fisher information for quantifying multipartite entanglement. This approach circumvents the need for analytical continuation and makes entanglement calculations feasible with a broad range of numerical methods. This measure is then applied to study the pseudogap phase in cuprates, a regime closely linked with superconductivity. Our findings indicate an increase in entanglement within the pseudogap phase, consistent with experimental observations.