Lead core rubber bearings are among the commonly used devices for seismic isolation of buildings and structures. They are a part of the so-called base isolation meant to decouple the superstructure from the ground, thus modifying the structure’s fundamental period and mitigating damage to structural and non-structural elements. Many insightful models have contributed to a better understanding of the lead core rubber bearings response considered as a separate unit or components of a seismically protected structure. However, the available models can still be enhanced. Furthermore, prediction of the dynamic structural response always offers, in most cases, unique problems; the incorporated models of the devices for seismic isolation play a crucial role in such a context. The article discusses an approach to predict the lead core rubber bearing response considered as a separate unit. To this end, the needed constitutive relations are defined on the material scale. The behavior of the isolator is then studied by transient finite element analysis. Both ready-to-use material models and original procedures (being developed) are considered. Additionally, some features of a data-driven algorithm designed to predict the response of a lead core rubber bearing are outlined.