A sleeve for a clutch release/engagement thrust bearing device. The sleeve provides a body with a first axial bearing surface configured to work together with a translationally mobile control member, a second axial bearing surface configured to interact with a non-rotating ring of a bearing, and a central bore around a central axis configured to accept a cylindrical guide. The central bore of the sleeve provides a first series of guide portions with cylindrical surfaces, a second series of guide portions with cylindrical surfaces, which are axially at the opposite end to the first series. The guide portions of the first series are offset with respect to the guide portions of the second series in the circumferential direction, and an intermediate bore portion that is recessed with respect to the guide portions. The intermediate bore portion extending axially between the first and second series of the guide portions.