  General relativity predicts that the rotational momentum flux of matter
twists the spacetime via a vector gravitomagnetic (frame-dragging) field, which
remains undetected in cosmology. This vector field induces an additional
gravitational lensing effect; at the same time, the momentum field sources the
kinetic Sunyaev-Zel'dovich (kSZ) effect. The common origin of these two effects
allows us to probe the gravitomagnetic signal via their cross-correlations. In
this paper, we explore the possibility of detecting the gravitomagnetic field
in $\Lambda$CDM by cross-correlating the weak-lensing convergence field with
the CMB temperature map, which is imprinted with the kSZ signal. This approach
allows us to extract the gravitomagnetic effect because the cross correlation
between the standard Newtonian contribution to the weak-lensing convergence
field, $\kappa_\Phi$, and the kSZ effect is expected to vanish. We study the
cross correlations with a suite of large-volume Newtonian $N$-body simulations
and a small-volume, high-resolution, general-relativistic counterpart. We show
that insufficient simulation resolution can introduce significant spurious
correlations between $\kappa_\Phi$ and kSZ. From the high-resolution
simulation, we find that the cumulative signal-to-noise ratio (SNR) of the
kSZ-gravitomagnetic convergence field can reach almost 15 (30) at
$\ell\simeq5000$ ($10^4$) for the lensing source redshift $z_s=0.83$, if only
cosmic variance is considered. We make forecast for next-generation lensing
surveys such as Euclid and LSST, and CMB experiments such as Simons Observatory
and CMB-S4, and find that, for $z_s=1.4$, the cumulative SNR can exceed 5 (9)
at $\ell\simeq5000$ ($10^4$), indicating that the cosmological gravitomagnetic
effect can be detected, if several foreground contaminations can be removed.
