We report on precise hyperspherical-basis calculations of ηNN and ηNNN quasibound states, using energy dependent ηN interaction potentials derived from coupled-channel models of the S11N⁎(1535) nucleon resonance. The ηN attraction generated in these models is too weak to generate a two-body bound state. No ηNN bound-state solution was found in our calculations in models where ReaηN≲1 fm, with aηN the ηN scattering length, covering thereby the majority of N⁎(1535) resonance models. A near-threshold ηNNN bound-state solution, with η separation energy of less than 1 MeV and width of about 15 MeV, was obtained in the 2005 Green–Wycech model where Re aηN≈1 fm. The role of handling self consistently the subthreshold ηN interaction is carefully studied.