A large fraction of observed protoplanetary disks in nearby Star-Forming Regions (SFRs) are fainter than expected in CO isotopologue emission. Disks not detected in 13CO line emission are also faint and often unresolved in the continuum emission at an angular resolution of around 0.2 arcseconds. Focusing on the Lupus SFR, the aim of this work is to investigate whether the population of CO-faint disks comprises radially extended and low mass disks - as commonly assumed so far - or if it is of intrinsically radially compact disks, an interpretation that we propose in this paper. The latter scenario was already proposed for individual sources or small samples of disks, while this work targets a large population of disks in a single SFR for which statistical arguments can be made. A new grid of physical-chemical models of compact disks has been run with DALI in order to cover a region of the parameter space that had not been explored before with this code. Such models have been compared with 12CO and 13CO ALMA observations of faint disks in Lupus. Disks that are not detected in 13CO emission and with faint or undetected 12CO emission are consistent with compact disk models. For radially compact disk, the emission of CO isotopologues is mostly optically thick and it scales with the surface area: i.e., it is fainter for smaller objects. The fraction of compact disks is potentially between roughly 50% and 60% of the entire Lupus sample. Deeper observations of 12CO and 13CO at a moderate angular resolution will allow us to distinguish whether faint disks are intrinsically compact, or if they are extended but faint, without the need of resolving them. If the fainter end of the disk population observed by ALMA disk surveys is consistent with such objects being very compact, this will either create a tension with viscous spreading or require MHD winds or external processes to truncate the disks.
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