We study the loop-induced flavor-changing decay $t\ensuremath{\rightarrow}c+\ensuremath{\gamma}$ in a nonlinear ${R}_{\ensuremath{\xi}}$ gauge for the standard model with three and four generations and in models with two Higgs doublets. We find that this decay could be relevant only in a scenario with two Higgs doublets and the mass of the fourth-generation ${b}^{\ensuremath{'}}$ quark larger than ${m}_{t}$. Under these conditions, the flavor-changing electromagnetic decays ${T}_{q}^{*}\ensuremath{\rightarrow}{D}_{q}\ensuremath{\gamma}({T}_{q}\ensuremath{\rightarrow}{D}_{q}^{*}\ensuremath{\gamma})$ may surpass the standard electromagnetic transition ${T}_{q}^{*}\ensuremath{\rightarrow}{T}_{q}\ensuremath{\gamma}$.