  We study, for the first time, the cross correlation between the angular
distribution of radial peculiar velocities (PV) and the lensing convergence of
cosmic microwave background (CMB) photons. We derive theoretical expectations
for the signal and its covariance and assess its detectability with existing
and forthcoming surveys. We find that such cross-correlations are expected to
improve constraints on different gravitational models by partially breaking
degeneracies with the matter density. We identify in the distance-scaling
dispersion of the peculiar velocities the most relevant source of noise in the
cross correlation. For this reason, we also study how the above picture changes
assuming a redshift-independent scatter for the PV, obtained for example using
a reconstruction technique. Our results show that the cross correlation might
be detected in the near future combining PV measurements from DESI and the
convergence map from CMB-S4. Using realistic direct PV measurements we predict
a cumulative signal-to-noise ratio of approximately $3.8 \sigma$ using data on
angular scales $3 \leq \ell \leq 200$. For an idealized reconstructed peculiar
velocity map extending up to redshift $z=0.15$ and a smoothing scale of $4$ Mpc
$h^{-1}$ we predict a cumulative signal-to-noise ratio of approximately $ 27
\sigma$ from angular scales $3 \leq \ell \leq200 $. We conclude that currently
reconstructed peculiar velocities have more constraining power than directly
observed ones, even though they are more cosmological-model dependent.
