In this work we examine a nonlinear version of the Jaynes-Cummings model for two identical two-level atoms allowing for Ising-like and dipole-dipole interplays between them. The model is said to be nonlinear in the sense that it can incorporate both a general intensity-dependent interaction between the atomic system and the cavity field and/or the presence of a nonlinear medium inside the cavity... As an example, we consider a particular type of atom-field coupling based upon the so-called Buck-Sukumar model and a lossless Kerr-like cavity. We describe the possible effects of such features on the evolution of some quantities of current interest, such as atomic excitation, purity, concurrence, the entropy of the field and the evolution of the latter in phase space. (read more)