  The position and movement of poles of the amplitude for elastic eta-meson
scattering off the light nuclei d, t, ^3He, and ^4He are studied. It is found
that, within the existing uncertainties for the elementary eta-N interaction,
all these nuclei can support a quasi-bound state. The values of the eta-nucleus
scattering lengths corresponding to the critical eta-N interaction which
produces a quasi--bound state are given.
