Abstract Coulomb breakup reactions of $^{93,94}$Zr have been studied in inverse kinematics at incident beam energies of about 200 MeV/nucleon in order to evaluate neutron capture reaction methods. The $^{93}$Zr(n,$\gamma$)$^{94}$Zr reaction is particularly important as a candidate nuclear transmutation reaction for the long-lived fission product $^{93}$Zr in nuclear power plants. One- and two-neutron removal cross sections on Pb and C targets were measured to deduce the inclusive Coulomb breakup cross sections, 375 $\pm$ 29 (stat.) $\pm$ 30 (syst.) and 403 $\pm$ 26 (stat.) $\pm$ 31 (syst.) mb for $^{93}$Zr and $^{94}$Zr, respectively. The results are compared with estimates using the standard Lorentzian model and microscopic calculations. The results reveal a possible contribution of the pygmy dipole resonance or giant quadrupole resonance in the Coulomb breakup reactions of $^{94}$Zr.