We discuss sensitivity of the $\gamma\gamma \to \tau^+\tau^-$ process in ultraperipheral Pb+Pb collisions at LHC energies on the anomalous magnetic moment of $\tau$ lepton ($a_\tau$). We derive the corresponding cross sections considering semileptonic decays of both leptons in the fiducial volume of ATLAS and CMS detectors. The expected limits on $a_\tau$ with the existing Pb+Pb dataset are found to be better by a factor of two comparing to current best experimental limits and can be further improved by another factor of two at High Luminosity LHC. In addition, our results for $\tau$ lepton electric dipole moment, $d_\tau$, can be competitive with the current best limits obtained by the Belle experiment.