We study the FCNC top quark decays t → c h in the framework of the MSSM, where h ≡ h0,H0,A0 is any of the supersymmetric neutral Higgs bosons. We include the leading set of SUSY-QCD and SUSY electroweak contributions. While the FCNC top quark decay into the SM Higgs boson has such a negligible rate that it will not be accessible to any presently conceivable accelerator, we find that there is a chance that the potential rates in the MSSM can be measured at the high luminosity colliders round the corner, especially at the LHC and possibly at a future LC, but we deem it difficult at the upgraded Tevatron. In view of the large SUSY-QCD effects that we find in the Higgs channels, and due to some discrepancies in the literature, we have revisited the FCNC top quark decay into gluon, t → c g, in our framework. We confirm that the possibility of sizeable rates does not necessarily require a general pattern of gluino-mediated FCNC interactions affecting both the LH and the RH sfermion sectors - the LH one being sufficient. However, given the present bounds on sparticle masses, the gluon channel turns out to lie just below the expected experimental sensibility, so our general conclusion is that the Higgs channels t → c h (especially the one for the light CP-even Higgs) have the largest potential top quark FCNC rates in the MSSM, namely of order 10-4. © 1999 Elsevier Science B.V. All rights reserved.