Magnetic Monopoles (MMs) are predicted in various theories of physics beyond the Standard Model. The introduction of MMs explains the electric charge quantisation and restores the symmetry in Maxwell's equations with respect to magnetic and electric fields. Despite intense experimental searches, they remain unobserved to date.
The Large Hadron Collider (LHC) achieves energies never reached before, opening possibilities for the discovery of exotic particles in the TeV mass range. MMs direct production is explored in the specialised MoEDAL experiment. Investigating the $\gamma \gamma \to \gamma \gamma$ process is worthwhile as this channel allows the study of possible contributions from new particles, such as MMs. In this paper, we investigate the observability of virtual MMs in this channel at the LHC, in both ultra-peripheral Pb-Pb and $pp$ collisions. Specifically, the reinterpretation of LHC Run 2 data has been performed in order to impose lower bounds on the MM mass. For this purpose, two different approaches have been used: Born-Infeld and Effective Field Theory.