We discuss a possibility of extracting anomalous electromagnetic couplings of the tau lepton in ultraperipheral ultrarelativistic heavy ion collisions. Nuclear cross section for exclusive $\tau^+ \tau^-$ production shows sizeable dependence on anomalous magnetic moment. We also investigated the expected sensitivity on $a_\tau$ and $d_\tau$, assuming standard LHC detectors using the currently available as well as future datasets. The expected limits on $a_{\tau}$ with existing Pb+Pb dataset are found to be better by a factor of two comparing to the current best experimental limits and can be further improved by another factor of two at High Luminosity LHC.