  A Higgs portal in the form of the operator $|H|^2$ provides a minimal and
theoretically motivated link between the Standard Model (SM) and new physics.
While Higgs portals can be constrained well by exotic Higgs decays if the
beyond-the-SM states are light, testing scenarios where these particles are
kinematically inaccessible is known to be challenging. We explore the
sensitivity of future hadron collider measurements of $ZZ \to 4 \ell$
production in constraining Higgs portal interactions. It is shown that by using
a matrix-element based kinematic discriminant the reach of the high-luminosity
option of the Large Hadron Collider (LHC) can be significantly enhanced
compared to studies that are based on measurements of the four-lepton invariant
mass spectrum alone. We also analyse the potential of the high-energy upgrade
of the LHC and a Future Circular Collider in constraining new physics that
couples to $|H|^2$. The obtained constraints are compared to the limits one
expects to find from other single-Higgs probes. In addition, we provide an
independent analysis of the relevant Higgs portal effects in double-Higgs
production. We find that the constraints obtained from our $ZZ \to 4 \ell$
analysis turn out to be both competitive with and complementary to the
projected limits obtained using other search techniques.
