We examine the prevalence of truncated star-forming discs in the Virgo cluster down to M* ≃ 107 M⊙. This work makes use of deep, high-resolution imaging in the Hα+[N II] narrow-band from the Virgo Environmental Survey Tracing Ionised Gas Emission (VESTIGE) and optical imaging from the Next Generation Virgo Survey (NGVS). To aid in the understanding of the effects of the cluster environment on star formation in Virgo galaxies, we take a physically motivated approach to define the edge of the star-forming disc via a drop-off in the radial specific star formation rate profile. A comparison with the expected sizes of normal galactic discs provides a measure of how truncated star-forming discs are in the cluster. We find that truncated star-forming discs are nearly ubiquitous across all regions of the Virgo cluster, including beyond the virial radius (0.974 Mpc). The majority of truncated discs at large cluster-centric radii are of galaxies likely on their first infall. As the intra-cluster medium density is low in this region, it is difficult to explain this population with solely ram-pressure stripping. A plausible explanation is that these galaxies are undergoing starvation of their gas supply before ram-pressure stripping becomes the dominant quenching mechanism. A simple model of starvation shows that this mechanism can produce moderate disc truncations within 1−2 Gyr. This model is consistent with ‘slow-then-rapid’ or ‘delayed-then-rapid’ quenching, whereby the early starvation mode drives disc truncations without significant change to the integrated star formation rate, and the later ram-pressure stripping mode rapidly quenches the galaxy. The origin of starvation may be in the group structures that exist around the main Virgo cluster, which indicates the importance of understanding pre-processing of galaxies beyond the cluster virial radius.