  The onset of binding $\eta$(548) mesons in nuclei is studied in the He
isotopes by doing precise $\eta NNN$ and $\eta NNNN$ few-body stochastic
variational method calculations for two semi-realistic $NN$ potentials and two
energy dependent $\eta N$ potentials derived from coupled-channel models of the
$N^{\ast}(1535)$ nucleon resonance. The energy dependence of the $\eta N$
subthreshold input is treated self consistently. It is found that a minimal
value of the real part of the $\eta N$ scattering length $a_{\eta N}$ close to
1 fm is required to bind $\eta$ mesons in $^3$He, yielding then a few MeV
$\eta$ binding in $^4$He. The onset of $\eta$-meson binding in $^4$He requires
that Re$\,a_{\eta N}$ exceeds 0.7 fm approximately. These results compare well
with results of recent $\eta NNN$ and $\eta NNNN$ pionless effective field
theory calculations. Related optical-model calculations are also discussed.
