Near-threshold scattering of eta-meson from light nuclei H-2, H-3, He-
3, and He-4 is considered on the basis of approximate few-body equatio
ns. The position and movement of poles of the elastic scattering ampli
tude have been studied. It was found that, within the existing uncerta
inties for the elementary eta N interaction, all these nuclei can supp
ort an eta-nucleus quasi-bound state. A steep increase of the eta(4)He
phase-shift by pi is found at a c.m. energy of similar to 10 MeV, whi
ch indicates that a near threshold resonance exists in this system.