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The rare top quark decays mediated by a new neutral massive gauge boson that is predicted in models with extended gauge symmetries are studied. We focus on the processes $t\to cV, uV$ induced at the one loop level, where $V =γ, g$, by considering different extended models. It is found that, within a broad range of mass of the new neutral gauge boson, the models predict branching ratios for the decays in study that are competitive with respect to the corresponding branching ratios in the standard model. In order to establish bound on our branching ratios, we consider the recent experimental bounds as $m_{Z^\prime}\geq$ 3.8-4.5 TeV, depending on the model, which also impose restrictions on our calculation. Even in this case, the resulting branching ratios are of the same order of magnitude as that predicted by the standard model. It should be noted that for the case of two models studied here, since no experimental bound exists to compare with, the results could be important, as they are, in the best of cases, two orders of magnitude larger than the predicted by the standard model.