We perform a complete and detailed analysis of the flavor changing neutral current rare top quark decays t → cl + l − and t → cν iνi at one-loop level in the Standard Model, Two Higgs Doublet Models (I and II) and in MSSM. The branching ratios are very small in all models, O(10 −14 ), except for the case of the unconstrained MSSM, where they can reach O(10 −6 ) for e + e − and ν iνi , and O(10 −5 ) for τ + τ − . This branching ratio is comparable to the ones for t → cV, cH. We also study the production rates of single top and the forward-backward asymmetry in e + e − → tc and comment on the observability of such a signal at the International Linear Collider.