We carry out a study in the approach of a dimension-eight effective field theory on the anomalous quartic WWγγ couplings that arise from the process pp → Wγγ at the Future Circular Collider-hadron hadron (FCC-hh). The process is sensitive to the WWγγ vertex, and future projections on the anomalous f M,i /Λ4 and f T,j /Λ4 couplings can be put by measurements of Wγγ production. We illustrate the physics opportunities and reach of the FCC-hh operating for a high integrated luminosity scenario of of proton–proton collisions at a center-of-mass energy of TeV and systematic uncertainties of δ sys = 0%, 3%, 5%, 10%. Production cross-sections of the signal and the sensitivity constraints on the anomalous f M,i /Λ4 and f T,j /Λ4 couplings at 95% C.L. are identified through the leptonic and hadronic decay modes of the W-boson. The FCC-hh sensitivity on these anomalous couplings is expected might reach up to the order of magnitude . These sensitivity perspectives indicate that the pp → Wγγ channel at the FCC-hh has a greater potential to search for the anomalous quartic WWγγ couplings and to search for new physics than present colliders.