We examine the potential of the process ep -> e gamma*gamma*p -> e tau(tau) over barp at the Large Hadron Electron Collider (LHeC) and the Future Circular Collider-h. dron electron (FCC-he) in order to examine non-standard tau(tau) over bar gamma coupling in a model-independent way by means of an effective Lagrangian approach. We perform pure leptonic and semileptonic decays for tau(-)tau(+) production in the final state. Furthermore, we use L = 10, 50, 100, 300, 500, 1000 fb(-1) with root s = 1.30, 1.98, 7.07, 10 TeV, and we consider systematic uncertainties of delta(sys) = 0, 5, 10%. The best sensitivity bounds obtained from the process ep -> e gamma*gamma*p -> e tau(tau) over barp on the anomalous couplings are -0.002 5 < <(a)over tilde>(tau) < 0.000 9 and vertical bar<(d)over tilde>(tau)vertical bar < 8.85 x 10(-18) e cm, respectively. Therefore, our results show that the process ep -> e gamma*gamma*p -> e tau(tau) over barp at the LHeC and FCC-he shows very good prospects for probing the anomalous magnetic and electric dipole moments of the tau lepton at the gamma*gamma* mode of the future ep collider.