Research related to food insecurity (FI) among European university student populations is currently limited, especially the studies carried out during the COVID-19 pandemic. The present study aimed to assess the prevalence and identify possible predictors of FI among students from a Spanish public university, the University of the Basque Country UPV/EHU, during the COVID-19 pandemic. A cross-sectional observational study design was used, in which a total of 422 students completed an online survey. Results were weighted according to age and field of education. Binary logistic regressions adjusted by sex, age, and campus were applied to identify FI predictors. FI in 19.6, 2.6, and 0.7% of the population was mild, moderate, and severe, respectively. The three main predictors of FI were a decrease in the main source of income (OR, 2.80; 95% IC, 2.57–3.06), not receiving scholarships during the pandemic (OR, 2.32; 95% IC, 2.18–2.47), and living arrangements before the pandemic (not living with parents/relatives) (OR, 2.03, 95% IC, 1.89–2.18). This study found a high prevalence of FI among the students surveyed, and the strongest predictors of this FI were related to socioeconomic status. A robust and comprehensive policy response is recommended to mitigate FI in this population.