  The strong energy dependence of the s-wave eta-N scattering amplitude at and
below threshold, as evident in coupled-channels K-matrix fits and chiral models
that incorporate the S11 N*(1535) resonance, is included self-consistently in
eta-nuclear bound-state calculations. This approach, applied recently in
calculations of kaonic atoms and Kbar-nuclear bound states, is found to impose
stronger constraints than ever on the onset of eta-nuclear binding, with a
minimum value of Re a_{eta N} approximately 0.9 fm required to accommodate an
eta-4He bound state. Binding energies and widths of eta-nuclear states are
calculated within several underlying eta-N models for nuclei across the
periodic table, including eta-25Mg for which some evidence was proposed in a
recent COSY experiment.
