General relativity predicts that the rotational momentum flux of matter twists the space–time 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 Λ cold dark matter by cross-correlating the weak-lensing convergence field with the cosmic microwave background (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, κΦ, 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 κΦ 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 ℓ ≃ 5000 (104) for the lensing source redshift zs = 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 zs = 1.4, the cumulative SNR can exceed 5 (9) at ℓ ≃ 5000 (104), indicating that the cosmological gravitomagnetic effect can be detected, if several foreground contaminations can be removed.