In this paper, a study is carried out on the $$e^-p \rightarrow e^-\gamma ^* p \rightarrow p W^-\gamma \nu _e$$
e
-
p
→
e
-
γ
∗
p
→
p
W
-
γ
ν
e
production to probe quartic $$W^+W^-\gamma \gamma $$
W
+
W
-
γ
γ
couplings at the Large Hadron electron Collider (LHeC) with $$\sqrt{s}= 1.30, 1.98\,\hbox {TeV}$$
s
=
1.30
,
1.98
TeV
and at the Future Circular Collider-hadron electron (FCC-he) with $$\sqrt{s}= 3.46, 5.29\,\hbox {TeV}$$
s
=
3.46
,
5.29
TeV
. Production cross-sections are determined for both at leptonic and hadronic decay channels of the W-boson. With the data from future $$e^-p$$
e
-
p
colliders, it is possible to obtain sensitivity measures at $$95\%$$
95
%
C.L. on the anomalous $$f_ {M,i}/\Lambda ^4$$
f
M
,
i
/
Λ
4
and $$ f_ {T,j}/\Lambda ^4$$
f
T
,
j
/
Λ
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 $$e^-p \rightarrow e^-\gamma ^* p \rightarrow p W^-\gamma \nu _e $$
e
-
p
→
e
-
γ
∗
p
→
p
W
-
γ
ν
e
in $$e^-p$$
e
-
p
collisions offers a window for study the quartic $$W^+W^-\gamma \gamma $$
W
+
W
-
γ
γ
electroweak bosons couplings at the LHeC and the FCC-he, which provides a much cleaner collision environment than the LHC.