Animal viruses frequently cause zoonotic disease in humans. As these viruses are highly diverse, evaluating the threat that they pose remains a major challenge, and efficient approaches are needed to rapidly predict virus-host compatibility. Here, we develop a combined computational and experimental approach to assess the compatibility of New World arenaviruses, endemic in rodents, with the host TfR1 entry receptors of different potential new host species. Using signatures of positive selection, we identify a small motif on rodent TfR1 that conveys species specificity to the entry of viruses into cells. However, we show that mutations in this region affect the entry of each arenavirus differently. For example, a human single nucleotide polymorphism (SNP) in this region, L212V, makes human TfR1 a weaker receptor for one arenavirus, Machupo virus, but a stronger receptor for two other arenaviruses, Junin and Sabia viruses. Collectively, these findings set the stage for potential evolutionary trade-offs, where natural selection for resistance to one virus may make humans or rodents susceptible to other arenavirus species. Given the complexity of this host-virus interplay, we propose a computational method to predict these interactions, based on homology modeling and computational docking of the virus-receptor protein-protein interaction. We demonstrate the utility of this model for Machupo virus, for which a suitable cocrystal structural template exists. Our model effectively predicts whether the TfR1 receptors of different species will be functional receptors for Machupo virus entry. Approaches such at this could provide a first step toward computationally predicting the "host jumping" potential of a virus into a new host species. We demonstrate how evolutionary trade-offs may exist in the dynamic evolutionary interplay between viruses and their hosts, where natural selection for resistance to one virus could make humans or rodents susceptible to other virus species. We present an algorithm that predicts which species have cell surface receptors that make them susceptible to Machupo virus, based on computational docking of protein structures. Few molecular models exist for predicting the risk of spillover of a particular animal virus into humans or new animal populations. Our results suggest that a combination of evolutionary analysis, structural modeling, and experimental verification may provide an efficient approach for screening and assessing the potential spillover risks of viruses circulating in animal populations.