  In this Letter we study the double diffractive vector meson production in
$e^+e^-$ collisions assuming the dominance of the BFKL pomeron exchange. We
consider the non-forward solution of the BFKL equation at high energy and large
momentum transfer and estimate the total cross section for the process $e^+e^-
\to e^+e^- V_1 V_2$ with antitagged $e^+$ and $e^-$, where $V_1$ and $V_2$ can
be any two vector mesons ($V_i = \rho, \omega, \phi, J/\Psi, \Upsilon$). The
event rates for the future linear colliders are given.
