We update the Standard Model predictions for the branching ratios of the rare top-quark decays t → cg(g), and evaluate the maximum rates that can be reached in the aligned as well as in the four conventional two-Higgs-doublet models (2HDMs) with Z 2 symmetries. Taking into account the relevant constraints on the model parameters from precision flavour observables, we find that the branching ratios of t → cg and t → cgg decays can reach up to 3.36 × 10 −9 and 2.95 × 10 −9 respectively, being therefore of the same order, in the aligned 2HDM (A2HDM). This is obviously different from the SM case, where the predicted branching ratio of the three-body decay t → cgg is about two orders of magnitude larger than that of the two-body decay t → cg. On the other hand, compared with the SM predictions, no significant enhancements are observed in the four conventional 2HDMs with Z 2 symmetries for the branching ratios of these two decays. Nevertheless, the predicted branching ratios of t → cg and t → cgg decays in the A2HDM are still out of the expected sensitivities of the future HL-LHC and FCC-hh.