  The production of a quarkonium -- pair in exclusive and diffractive processes
in $pp$ collisions at the LHC and FCC energies is investigated. We consider the
$J/\Psi J/\Psi$ and $\Upsilon \Upsilon$ production in these processes and
present predictions for the transverse momentum and rapidity distributions
considering the kinematical ranges expected to be covered by central and
forward detectors. Results for the cross sections are also presented. Our
results indicate that the double $J/\Psi$ production is dominated by the
exclusive process, while the double $\Upsilon$ production receive a large
contribution of the diffractive process. The impact of the modelling of the gap
survival factor on our predictions is discussed.
