This Letter presents a search for highly ionizing magnetic monopoles in 262 μb−1 of ultraperipheral Pb+Pb collision data at sNN=5.36 TeV collected by the ATLAS detector at the LHC. A new methodology that exploits the properties of clusters of hits reconstructed in the innermost silicon detector layers is introduced to study highly ionizing particles in heavy-ion data. No significant excess above the background, which is estimated using a data-driven technique, is observed. Using a nonperturbative semiclassical model, upper limits at 95% confidence level are set on the cross section for pair production of monopoles with a single Dirac magnetic charge in the mass range of 20–150 GeV. Depending on the model, monopoles with a single Dirac magnetic charge and mass below 80–120 GeV are excluded.