  We study the indirect constraints on the quartic Higgs self-coupling that
arise from double-Higgs production at future hadron colliders. To this purpose,
we calculate the two-loop contributions to the $gg \to hh$ amplitudes that
involve a modified $h^4$ vertex. Based on our results, we estimate the reach of
a $pp$ collider operating at $27 \, {\rm TeV}$ and $100 \, {\rm TeV}$
centre-of-mass energy in constraining the cubic and quartic Higgs
self-couplings by measurements of double-Higgs and triple-Higgs production in
gluon-fusion.
