  A sensitivity of the VEPP-2000 $e^+e^-$ collider in a search for the rare
decay $\eta \rightarrow e^+ e^-$ has been studied. The inverse reaction $e^+
e^- \rightarrow \eta$ is proposed for this search. We have analyzed a data
sample with an integrated luminosity of 108 nb$^{-1}$ collected with the SND
detector in the center-of-mass energy range 520-580 MeV and found no background
events for the reaction $e^+ e^- \rightarrow \eta$ in the decay mode
$\eta\to\pi^0\pi^0\pi^0$. In the absence of background, a sensitivity to ${\cal
B}(\eta \rightarrow e^+ e^-)$ of $10^{-6}$ can be reached during two weeks of
VEPP-2000 operation. Such a sensitivity is better than the current upper limit
on ${\cal B}(\eta \rightarrow e^+ e^-)$ by a factor of 2.3.
