  The Higgs self-interactions play a crucial role for exploring the underlying
mechanism of electroweak symmetry breaking and the nature of the phase
transition involved. In this article, we propose to probe the quartic Higgs
self-interaction at lepton and hadron colliders, via the di-Higgs productions.
We analyze the contributions of the quartic Higgs coupling, including the
renormalization of the cubic Higgs coupling and the modification of the $VVhh$
form factor, to the vector-boson-fusion and the vector-boson associated
di-Higgs productions at one-loop level. Such an effect is independent of the
choice of gauge-fixing, if the quartic Higgs coupling is decoupled from the
other ones in the contexts considered. Notably, a combination of these two
di-Higgs productions is important for optimizing the collider sensitivities to
probe the quartic Higgs coupling. With this guideline, we explore the ILC and
CLIC sensitivities, and find that the ILC has a potential to measure the
quartic Higgs coupling, normalized by its SM value, with a precision of $\sim
\pm 25$ (500 GeV, 4 ab$^{-1}$ + 1 TeV, 2.5 ab$^{-1}$) and $\sim \pm20$ (500
GeV, 4 ab$^{-1}$ + 1 TeV, 8 ab$^{-1}$), at $1\sigma$ C.L., after marginalizing
the cubic Higgs coupling in the $\chi^2$ analysis. The dependence on the
renormalization scheme of the cubic Higgs coupling is discussed.
