  We discuss the formalism of two Higgs doublet model type III with CP
violation from CP-even CP-odd mixing in the neutral Higgs bosons. The flavor
changing interactions among neutral Higgs bosons and fermions are presented at
tree level in this type of model. These assumptions allow the study rare top
decays mediated by neutral Higgs bosons, particularly we are interested in
$t\rightarrow c l^+l^-$. For this process we estimated upper bounds of the
branching ratios $\textrm{Br}(t\rightarrow c \tau^+\tau^-)$ of the order of
$10^{-9}\sim 10^{-7}$ for a neutral Higgs boson mass of 125 GeV and
$\tan\beta=1$, 1.5, 2, 2.5. For the case of $t\rightarrow c \tau^+\tau^-$ the
number of possible events is estimated from 1 to 10 events which could be
observed in future experiments at LHC with a luminosity of 300
$\textrm{fb}^{-1}$ and 14 GeV for the energy of the center of mass. Also we
estimate that the number of events for the process $t\rightarrow c l^+l^-$ in
different scenarios is of order of $2500$.
