The KM3NeT research infrastructure is under construction in the Mediterranean Sea. It consists of two water Cherenkov neutrino detectors, ARCA and ORCA, aimed at neutrino astrophysics and oscillation research, respectively. Instrumenting a large volume of sea water with $sim$ 6,200 optical modules comprising a total of $sim$ 200,000 photomultiplier tubes, KM3NeT will achieve sensitivity to $sim$ 10 MeV neutrinos from Galactic and near-Galactic core-collapse supernovae through the observation of coincident hits in photomultipliers above the background. In this paper, the sensitivity of KM3NeT to a supernova explosion is estimated from detailed analyses of background data from the first KM3NeT detection units and simulations of the neutrino signal. The KM3NeT observational horizon (for a $5,sigma$ discovery) covers essentially the Milky-Way and for the most optimistic model, extends to the Small Magellanic Cloud ($sim$ 60 kpc). Detailed studies of the time profile of the neutrino signal allow assessment of the KM3NeT capability to determine the arrival time of the neutrino burst with a few milliseconds precision for sources up to 5$-$8 kpc away, and detecting the peculiar signature of the standing accretion shock instability if the core-collapse supernova explosion happens closer than 3$-$5 kpc, depending on the progenitor mass. KM3NeTs capability to measure the neutrino flux spectral parameters is also presented.
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