We predict a nonthermal magneto-optical effect for magnetic insulators subject to intense light carrying orbital angular momentum (OAM). Using a classical approach to second harmonic generation in nonlinear media with specific symmetry properties we predict a significant nonlinear contribution to the local magnetic field triggered by light with OAM. The resulting magnetic field originates from the displacement of electrons driven by the electrical field (with amplitude E₀) of the spatially inhomogeneous optical pulse, modeled here as a Laguerre-Gaussian beam carrying OAM. In particular, the symmetry properties of the irradiated magnet allow for magnetic field responses which are second-order ( ∼  E²₀) and fourth-order ( ∼  E⁴₀) in electric-field strength and have opposite signs. For sufficiently high laser intensities, terms  ∼  E⁴₀ dominate and generate magnetic field strengths which can be as large as several Tesla. Moreover, changing the OAM of the laser beam is shown to determine the direction of the total light-induced magnetic field, which is further utilized to study theoretically the nonthermal magnetization dynamics.