  We present a study of central exclusive production of $W^+W^-$ pairs in
proton-proton collisions at the LHC. We compare the contribution of the $\gamma
\gamma \to W^+ W^-$ mechanism with a new mechanism of exclusive diffractive
production through the $gg \to W^+ W^-$ subprocess with intermediate virtual
Higgs boson and quark box diagrams. The amplitude for the latter process is
expressed in terms of the off-diagonal unintegrated gluon distribution
functions. Several observables related to this process are calculated. The
phase space integrated diffractive contribution when separated is only a small
fraction of fb compared to 115.4 fb of the $\gamma\gamma$-contribution without
absorption. This opens a possibility of efficient searches for anomalous boson
$\gamma W^+ W^-$ and $\gamma \gamma W^+ W^-$ couplings due to new physics
beyond Standard Model.
