Massive high-redshift quiescent compact galaxies (nicknamed red nuggets) have been traditionally connected to present-day elliptical galaxies, often overlooking the relationships that they may have with other galaxy types. We use large bulge-disc decomposition catalogues based on the Sloan Digital Sky Survey to check the hypothesis that red nuggets have survived as compact cores embedded inside the haloes or discs of present-day massive galaxies. In this study, we designate a compact core as the bulge component that satisfies a prescribed compactness criterion. Photometric and dynamic mass-size and mass-density relations are used to show that, in the inner regions of galaxies at z ˜ 0.1, there are abundant compact cores matching the peculiar properties of the red nuggets, an abundance comparable to that of red nuggets at z ˜ 1.5. Furthermore, the morphology distribution of the present-day galaxies hosting compact cores is used to demonstrate that, in addition to the standard channel connecting red nuggets with elliptical galaxies, a comparable fraction of red nuggets might have ended up embedded in discs. This result generalizes the inside-out formation scenario; present-day massive galaxies can begin as dense spheroidal cores (red nuggets), around which either a spheroidal halo or a disc is formed later.