Stars emit megaelectronvolt neutrinos during their evolution via nuclear syntheses and thermal processes, and detecting them could provide insights into stellar structure beyond what is accessible through electromagnetic wave observations. So far, megaelectronvolt neutrinos have been observed from the Sun and SN 1987A. It has been suggested that pre-supernova stars in the oxygen- and silicon-burning stages would emit enough megaelectronvolt neutrinos to be detectable on Earth, provided they are in the local Universe. In this study, we investigate the prospect of detecting neutrinos from red supergiants (RSGs) in the carbon-burning phase. In our Galaxy, around a thousand RSGs have been cataloged, and several are expected to be in the carbon-burning phase. We first calculate the luminosity and energy spectrum of the neutrinos emitted during the post-main-sequence evolution of massive stars. For a nearby carbon-burning RSG located similar to 200 pc away, we estimate the neutrino flux reaching Earth to be as large as similar to 105 cm-2 s-1, with a spectrum peaking at similar to 0.6 MeV. We then assess the feasibility of detecting these neutrinos in underground facilities, particularly in hybrid detectors equipped with a water-based liquid scintillator and ultrafast photodetectors. In detectors with a volume comparable to Super-Kamiokande, for the above flux, we anticipate up to similar to 50 neutrino events per year with directional information. Although this is a fair number, the number of events from radioactive backgrounds would be much larger. Our results indicate that studying neutrinos from carbon-burning RSGs and predicting supernovae well in advance before their explosion would be challenging with currently available detector technologies.