The pure squeeze magneto-elastohydrodynamic lubrication (MEHL) motion of circular contacts with an electrically conducting fluid in the presence of a transverse magnetic field is explored under constant load condition. The differences between classical elastohydrodynamic lubrication and MEHL are discussed. The results reveal that the effect of an externally applied magnetic field is equivalent to enhancing effective lubricant viscosity. Therefore, as the Hartmann number increases, the enhancing effect becomes more obvious. Furthermore, the transient pressure profiles, film shapes, normal squeeze velocities, and effective viscosity during the pure squeeze process under various operating conditions are discussed.