We study possible non-standard $tuZ$ and $tcZ$ couplings, which induce flavor-changing neutral-current decays of the top quark, in the effective-Lagrangian framework. The corresponding interaction Lagrangian comes from several $SU(3) \times SU(2) \times U(1)$ invariant dimension-6 effective operators, and it includes four independent complex-number coupling constants. Constraints on those non-standard coupling constants in each interaction are derived by using the present experimental limits of the branching fractions for $t \to u Z$ and $t \to c Z$ processes. Expected improvements of the constraints at future facilities are discussed as well. It is also pointed out that some correlations hold among those constrained coupling constants.