We conducted a study on measuring W + W − production and on the sensitivity limits at 95% Confidence Level on thirteen anomalous couplings obtained by dimension-8 operators which are related to the anomalous quartic W W γγ couplings. We consider the main e − p → e − γ * γ * p → e − W + W − p reaction with the sub-process γ * γ * → W + W − at the Large Hadron electron Collider (LHeC) and the Future Circular Collider-hadron electron (FCC-he). For the LHeC, energies of the e − beams are taken to be E e = 60 and 140 GeV and the energy of the p beams is taken to be E p = 7 TeV. For the FCC-he, energies of the e − beams are taken to be E e = 60 and 140 GeV and the energy of the p beams is taken to be E p = 50 TeV, respectively. It is interesting to notice that the LHeC and the FCC-he will lead to model-independent limits on the anomalous quartic W W γγ couplings which are one order of magnitude stringent than the CMS Collaboration limits, in addition to being competitive with other limits reported in the literature. PACS numbers: 12.60.-i, 14.70.Fm, 4.70.Bh Keywords: Models beyond the standard model, W bosons, Quartic gauge boson couplings.