High-intensity X-ray sources are essential diagnostic tools for science, technology and medicine. Such X-ray sources can be produced in laser-plasma accelerators, where electrons emit short-wavelength radiation due to their betatron oscillations in the plasma wake of a laser pulse. Contemporary available betatron radiation X-ray sources can deliver a collimated X-ray pulse of duration on the order of several femtoseconds from a source size of the order of several micrometres. In this paper we demonstrate, through particle-in-cell simulations, that the temporal resolution of such a source can be enhanced by an order of magnitude by a spatial modulation of the emitting relativistic electron bunch. The modulation is achieved by the interaction of the that electron bunch with a co-propagating laser beam which results in the generation of a train of equidistant sub-femtosecond X-ray pulses. The distance between the single pulses of a train is tuned by the wavelength of the modulation laser pulse. The modelled experimental setup is achievable with current technologies. Potential applications include stroboscopic sampling of ultrafast fundamental processes.