Future liquid-argon DarkSide-20k and ARGO detectors, designed for direct dark matter search, will be sensitive also to core-collapse supernova neutrinos, via coherent elastic neutrino-nucleus scattering. This interaction channel is flavor-insensitive with a high-cross section, enabling for a high-statistics neutrino detection with target masses of $\sim$50~t and $\sim$360~t for DarkSide-20k and ARGO, respectively. Thanks to the low-energy threshold of $\sim$0.5~keV$_{nr}$ achievable by exploiting the ionization channel, DarkSide-20k and ARGO have the potential to discover supernova bursts throughout our galaxy and up to the Small Magellanic Cloud, respectively, assuming a 11-M$_{\odot}$ progenitor star. We report also on the sensitivity to the neutronization burst, whose electron neutrino flux is suppressed by oscillations when detected via charged current and elastic scattering. Finally, the accuracies in the reconstruction of the average and total neutrino energy in the different phases of the supernova burst, as well as its time profile, are also discussed, taking into account the expected background and the detector response.