  Decays of the top quark induced by flavor changing neutral currents (FCNC)
are known to be extremely rare events within the Standard Model. This is so not
only for the decay modes into gauge bosons, but most notably in the case of the
Higgs channels, e.g. t-> H_{SM}+c, with a branching fraction of 10^{-13} at
most. Therefore, detection of FCNC top quark decays in a future high-energy,
and high-luminosity, machine like the LHC or the LC would be an indisputable
signal of new physics. In this paper we show that within the simplest extension
of the SM, namely the general two-Higgs-doublet model, the FCNC top quark
decays into Higgs bosons, t->(h^{0},H^{0},A^{0})+c, can be the most favored
FCNC modes -- comparable or even more efficient than the gluon channel t-> g+c.
In both cases the optimal results are obtained for Type II models. However,
only the Higgs channels can have rates reaching the detectable level (10^{-5}),
with a maximum of order 10^{-4} which is compatible with the charged Higgs
bounds from radiative B-meson decays. We compare with the previous results
obtained in the Higgs sector of the MSSM.
