  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. KM3NeT's
capability to measure the neutrino flux spectral parameters is also presented.
