The energies and widths of bound states of the eta meson in different nuclei are obtained using the results for its self-energy in a nuclear medium, which is evaluated in a self-consistent manner using techniques of unitarized chiral perturbation theory. We find bound states in all studied nuclei (from C-12 on) and the half widths obtained are larger than the separation of the levels, what makes the experimental observation of peaks unlikely. We have paid a special attention to the region of nuclei where only the 1s state appears and the binding energies are of the order of magnitude of the half width, which would magnify the chances that some broad peak could be observed. This is found in the region of Mg-24 with a binding energy around 12.6 MeV and half width of 16.7 Mev. In heavy nuclei like Pb-208 there are many bound states which would be difficult to disentangle and the deepest state has a binding energy about 21 MeV and half width around 16 Mev. Such an overlapping accumulation of states could be seen as an extension of the continuum of eta strength into the bound region in eta production experiments.