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Magnetohydrodynamics (MHD) investigates the intricate relationship between electromagnetism and fluid dynamics, offering a complete insight into the behavior of conducting fluids under the influence of magnetic fields. This theory plays a pivotal role in the framework of magnetic confinement fusion, where it can be applied to describe both thermonuclear plasmas confined inside the vacuum vessel and operating fluids, such as liquid metals and molten salts, flowing within the blanket of future tokamaks. Currently, the state-of-the-art numerical modeling of MHD scenarios employs a multi-physics framework to examine the interplay between magnetic fields and thermal hydraulics; however, due to the complexity of the involved physics, detailed models are required, resulting in a significant computational burden. In this regard, reduced order modeling (ROM) techniques may represent a promising solution, as they enable approximating complex systems with lower-dimensional models. Indeed, ROM methodologies can significantly reduce the required computational time while maintaining accuracy in capturing the convoluted physics involved in fusion reactors, especially in the contexts of sensitivity analysis, uncertainty quantification, and control. Despite their potential, ROM methods are relatively under-explored within the MHD framework; this study applies ROM techniques to MHD scenarios, focusing on their capabilities and possible limitations. To this aim, the backward-facing step, which is well suited for exploring the effects of different magnetic fields on turbulent dynamics, is adopted as case study. In particular, this work evaluates the potentialities of the ROM approach in enhancing computational efficiency within the MHD domain. Each of the methods evaluated was effective in precisely reconstructing flow dynamics at any given time and across the full range of magnetic field values tested while significantly reducing computational costs compared to full-order simulations. Practically, this study demonstrates the feasibility to create simplified models that accurately represent the magnetohydrodynamic flows of fluids within the blanket. [ABSTRACT FROM AUTHOR]