We explore the physics expected sensitivity at the Large Hadron electron Collider (LHeC) and the future circular collider-hadron electron (FCC-he) to search for the anomalous quartic WWγγ couplings in single W-boson production in association with a photon. We study the process ep → e−γ*p → eWγq′X via the subprocess γ*q → Wγq′. The center-of-mass energies and luminosities of the LHeC are assumed to be TeV, and for the FCC-he TeV and . Considering these energies and luminosities, we estimate sensitivity measures on the anomalous quartic WWγγ couplings at 95% C.L., which can be an order of magnitude more stringent than the experimental limits reported by ATLAS and CMS Collaborations at the LHC.