Abstract We present a phenomenology study on central exclusive production of W+W− boson pairs in proton-proton collisions at the Large Hadron Collider at 14 TeV using the forward proton detectors, such as the ATLAS Forward Proton or the CMS-TOTEM Precision Proton Spectrometer detectors. Final states where at least one of the W bosons decay hadronically in a large-radius jet are considered. The latter extends previous efforts that consider solely leptonic final states. A measurement of exclusive W+W− also allows us to further constrain anomalous quartic gauge boson interactions between photons and W bosons. Expected limits on anomalous quartic gauge couplings $$ {a}_{0,C}^W $$ a 0 , C W associated to dimension-six effective operators are derived for the hadronic, semi-leptonic, and leptonic final states. It is found that the couplings can be probed down to one-dimensional values of $$ {a}_0^W=3.7\times {10}^{-7}{\mathrm{GeV}}^{-2} $$ a 0 W = 3.7 × 10 − 7 GeV − 2 and $$ {a}_C^W=9.2\times {10}^{-7}{\mathrm{GeV}}^{-7} $$ a C W = 9.2 × 10 − 7 GeV − 7 at 95% CL at an integrated luminosity of 300 fb−1 by combining all final states, compared to values of about $$ {a}_0^W=4\times {10}^{-6}{\mathrm{GeV}}^{-2} $$ a 0 W = 4 × 10 − 6 GeV − 2 and $$ {a}_C^W=1\times {10}^{-5}{\mathrm{GeV}}^{-2} $$ a C W = 1 × 10 − 5 GeV − 2 at 95% CL expected for the leptonic channel alone.