The position and movement of poles of the amplitude for elastic eta-me
son scattering off the light nuclei H-2, H-3, He-3, and He-4 are studi
ed. It is found that, within the existing uncertainties for the elemen
tary eta N interaction, all these nuclei can support a quasibound stat
e. The values of the eta-nucleus scattering lengths corresponding to t
he critical eta N interaction that produces a quasibound state are giv
en.