The detection of a neutrino burst from the next Galactic Core-Collapse Supernova (CCSN) will
provide us invaluable information on this extreme phenomenon. Furthermore, the detection of its
gravitational waves and electromagnetic signals would give us a complete picture of all emitted
messengers. KM3NeT is a neutrino telescope consisting of two detectors, ORCA and ARCA,
currently under deployment in the Mediterranean Sea. By looking for an excess of coincidence
events above the optical background, it will be able to detect low-energy neutrinos from CCSN. A
sensitivity to Galactic and near-Galactic events is expected when data from the two infrastructures
is combined. With its integration in the SNEWS global alert network and the ongoing work to
compute and combine the neutrino light-curves of different detectors, KM3NeT will play a key
part in notifying other telescopes before the arrival of the other messengers. In this contribution,
we present the real-time detection capabilities of KM3NeT, the additional information that can be
brought by light-curve computations and the follow-up of external alerts.