  Motivated by the two recent observations in the WASA-at-COSY detector, we
investigate the $\eta^3$He nucleus with the $\eta NNN$ few body method. We
construct the effective $s$-wave energy dependent $\eta N$ potential which
reproduce the $\eta N$ subthreshold scattering amplitude in the 2005
Green-Wycech model. It gives the $\eta$ separation energy and decay width of
0.19 MeV and 1.71 MeV, respectively. We also construct various sets of
effective $s$-wave energy independent $\eta N$ potentials where the
corresponding complex scattering lengths (a) are within the range given in most
theoretical models. We obtain the bound $\eta^3$He nucleus with decay width of
about 5 MeV when a is (1.0 fm, 0.3 fm), and of about 10 MeV when a is (1.0 fm,
0.5 fm).
