In the absence of ongoing star formation, the light emitted by stars in a galaxy becomes progressively redder and fainter, which can potentially alter the morphological appearance of the galaxy itself.  In this study, we investigate how the ageing of stellar populations can drive a morphological transformation in galaxies whose star formation (SF) activity has been quenched on short timescales, like in cluster galaxies subject to ram pressure stripping from the intracluster medium. For this purpose, we use a sample of 91 galaxies with MUSE data from the GASP program and of their spatially resolved SF history derived with the spectral modelling software SINOPSIS. We simulate the future continuation of the SF activities by exploring two quenching scenarios: an instantaneous truncation of the SF across the whole disc, and an outside-in quenching with typical stripping timescales of 0.5 Gyr and 1 Gyr. For each scenario we produce mock MUSE spectroscopic datacubes and optical images for our galaxies during their evolution, and classify their morphology using an ad-hoc diagnostic tool. We find that, in all scenarios considered, the initial galaxy population dominated by blue-cloud spirals (∼ 90%) evolves into a mixed population mostly composed by red-sequence spirals (50 − 55%) and lenticulars (∼ 40%). The morphology transformation is completed after just 1.5 − 3.5 Gyr, proceeding faster in more efficient quenching scenarios. Our results indicate that, even without accounting for dynamical processes, SF quenching caused by the harsh environment of a cluster can significantly affect the morphology of the infalling galaxy population on timescales of a few Gyr.