The KM3NeT neutrino telescope is currently being deployed in the Mediterranean Sea. While the two sites, ORCA designed to study sub-TeV neutrinos and ARCA optimized for the TeV-PeV range, are not yet completed, the current effective area allows us to carry out the first searches for astrophysical neutrinos. In this contribution, we present the results of the search for MeV-TeV neutrinos from compact binary mergers detected during the third observing run of the LIGO and Virgo interferometers. We present independent searches focusing on the MeV and GeV-TeV ranges, as well as the results of a stacking analysis probing neutrino emission in different populations of mergers. This contribution will also present the prospects for similar searches with KM3NeT in a near future.