We study the decay of tangential velocity profiles with distance from a local disturbance in hard-sphere colloidal suspensions as the colloidal glass transition is approached. The disturbance, generated by a dimer of superparamagnetic particles rotated by an external magnetic field, enables a precise characterization of the system's response through confocal microscopy and tracking of individual particle dynamics. The tangential velocity profiles exhibit nearly exponential decay with distance. As particle density increases toward the colloidal glass transition, the characteristic length scale derived from exponential fits grows. We also observe that the colloidal particles slip against the rotating dimer, with less slip in samples which are closer to the glass transition.