Abstract We model the late evolution and mass loss history of rapidly rotating Wolf–Rayet stars in the mass range 5 M ⊙…100 M ⊙). We find that quasi-chemically homogeneously evolving single stars computed with enhanced mixing retain very little or no helium and are compatible with Type Ic supernovae. The more efficient removal of core angular momentum and the expected smaller compact object mass in our lower-mass models lead to core spins in the range suggested for magnetar-driven superluminous supernovae. Our higher-mass models retain larger specific core angular momenta, expected for long-duration gamma-ray bursts in the collapsar scenario. Due to the absence of a significant He envelope, the rapidly increasing neutrino emission after core helium exhaustion leads to an accelerated contraction of the whole star, inducing a strong spin-up and centrifugally driven mass loss at rates of up to in the last years to decades before core collapse. Because the angular momentum transport in our lower-mass models enhances the envelope spin-up, they show the largest relative amounts of centrifugally enforced mass loss, i.e., up to 25% of the expected ejecta mass. Our most massive models evolve into the pulsational pair-instability regime. We would thus expect signatures of interaction with a C/O-rich circumstellar medium for Type Ic superluminous supernovae with ejecta masses below ∼10 M ⊙ as well as for the most massive engine-driven explosions with ejecta masses above ∼30 M ⊙. Signs of such interaction should be observable at early epochs of the supernova explosion; they may be related to bumps observed in the light curves of superluminous supernovae, or to the massive circumstellar CO-shell proposed for Type Ic superluminous supernova Gaia16apd.