Context. The collapsar model requires rapidly rotating Wolf-Rayet stars as progenitors of long gamma-ray bursts. However, Galactic Wolf-Rayet stars rapidly lose angular momentum due to their intense stellar winds. Aims. We investigate whether the tidal interaction of a Wolf-Rayet star with a compact object in a binary system can spin up the Wolf-Rayet star enough to produce a collapsar. Methods. We compute the evolution of close Wolf-Rayet binaries, including tidal angular momentum exchange, differential rotation of the Wolf-Rayet star, internal magnetic fields, stellar wind mass loss, and mass transfer. The Wolf-Rayet companion is approximated as a point mass. We then employ a population synthesis code to infer the occurrence rates of the various relevant binary evolution channels. Results. We find that the simple scenario – i.e., the Wolf-Rayet star being tidally spun up and producing a collapsar – does not occur at solar metallicity and may only occur with low probability at low metallicity. It is limited by the widening of the binary orbit induced by the strong Wolf-Rayet wind or by the radius evolution of the Wolf-Rayet star that most often leads to a binary merger. The tidal effects enhance the merger rate of Wolf-Rayet stars with black holes such that it becomes comparable to the occurrence rate of long gamma-ray bursts.