  We suggest that the $\gamma p \to \phi \eta p$ reaction cross section, in the
kinematics where the $\eta p$ invariant mass in the final state lies between
the threshold value (m$_p$+m$_\eta$) and the N*(1535) resonance mass, is
largely determined by the $\eta N$ scattering amplitude close to threshold. The
initial photon energy is chosen in the range $4 < E_\gamma^{Lab} < 5$ GeV, in
order to reach low (absolute) values of the squared 4-momentum transfer from
the initial photon to the final $\phi$-meson. In these conditions, we expect
the t-channel $\pi^0$- and $\eta$-meson exchanges to drive the dynamics
underlying the $\gamma p \to \phi \eta p$ process. We show that the
$\eta$-exchange is the dominating contribution to the cross section while the
$\pi^0$-exchange is negligible. The $\eta$-$\pi^0$ interference is of the order
of $20 - 30 %$. The sign of this term is not known and alters significantly our
results. Data on the $\gamma p \to \phi \eta p$ process would be therefore very
useful to help unravelling the behavior of the $\eta p$ scattering amplitude
close to threshold and assessing the possibility of producing $\eta$-nucleus
bound states.
