The detection of a neutrino burst from the next Galactic Core-Collapse SuperNova (CCSN) will yield major breakthroughs in different fields of fundamental physics. The KM3NeT ORCA and ARCA neutrino telescopes in the Mediterranean Sea are expected to observe a significant number of neutrino interactions through the detection of Cherenkov light mostly induced by inverse beta decay processes in sea water. The detection of photons in coincidence between the 31 photomultipliers of KM3NeT digital optical modules (DOMs) allows to discriminate the CCSN signal against radioactive decays, bioluminescence and atmospheric muon backgrounds.The detection sensitivity and the potential to resolve the time profile of the neutrinos have been studied by means of accurate Monte Carlo simulations coupled to in-depth studies of the KM3NeT background features. Online triggering criteria have been determined to send real-time alerts for low-energy CCSN neutrino bursts, and also to meet the requirements for the integration in the SNEWS global alert network. In view of future developments, a preliminary study has been conducted on the determination of the neutrino arrival time and the fast triangulation of the source by different detectors to constrain the CCSN localisation in the sky.