  A study of the $\eta$-3N interaction in the energy region near the $\eta$-^3H
elastic scattering threshold is presented. The calculational scheme is based on
the four-body scattering formalism. A manageable form of the integral equations
is achieved by using a separable ansatz for the driving two-body $\eta N$- and
NN-forces as well as for the subamplitudes appearing in the (1+3) and (2+2)
partitions of the $\eta$-3N system. Results presented for the $\eta$-^3H
scattering length point to the existence of a virtual (antibound) s-wave
$\eta$-^3H state. As a consequence, a large enhancement of the cross section
for $\eta$-production from three-body nuclei is predicted.
