  In this paper, a study is carried out on the $e^-p \to e^-\gamma^* p \to p
W^+\gamma \nu_e$ production to probe quartic $W^+W^-\gamma\gamma$ couplings
using 10, 100 ${\rm fb^{-1}}$ of $e^-p$ collisions data at $\sqrt{s}$= 1.30,
1.98 GeV at the Large Hadron electron Collider (LHeC) and 100, 1000 ${\rm
fb^{-1}}$ with $\sqrt{s}$= 3.46, 5.29 GeV at the Future Circular
Collider-hadron electron (FCC-he). Production cross-sections are determined for
both at leptonic and hadronic decay channel of the $W$-boson. With the data
from future $e^-p$ colliders, it is possible to obtain sensitivity measures at
$95\%$ C.L. on the anomalous $f_ {M,i}/\Lambda^4$ and $ f_ {T,i}/\Lambda^4$
couplings which are competitive with the limits obtained by the LHC, as well as
with others limits reported in the literature. The production mode $e^-p \to
e^-\gamma^* p \to p W^+\gamma \nu_e $ in $e^-p$ collisions offers a window for
study the quartic $W^+W^-\gamma\gamma$ electroweak bosons couplings at the LHeC
and the FCC-he, which provides a much cleaner collision environment than the
LHC.
