  The production of fully-heavy tetraquark states in proton-proton ($pp$) and
proton-nucleus ($pA$) collisions at the center-of-mass energies of the Large
Hadron Collider (LHC) and at the Future Circular Collider (FCC) is investigated
considering that these states are produced through the double parton scattering
mechanism. We estimate the cross sections for the $T_{4c}$, $T_{4b}$ and
$T_{2b2c}$ states and present predictions for $pp$, $pCa$ and $pPb$ collisions
considering the rapidity ranges covered by central and forward detectors. We
demonstrate that the cross sections for $pA$ collisions are enhanced in
comparison to the $pp$ predictions scaled by the atomic number. Moreover, our
results indicate that a search of these exotic states is, in principle,
feasible in the future runs of the LHC and FCC.
