We report a magnetic study on nanostructured (Fe79Mn21)1xCux (0.00x0.30) alloys using static magnetic measurements. The alloys are mainly composed by an antiferromagnetic fcc phase and a disordered region that displays a spin-glass-like behavior. The interplay between the antiferromagnetic and magnetically disordered phases establishes an exchange anisotropy that gives rise to a loop shift at temperatures below the freezing temperature of moments belonging to the disordered region. The loop shift is more noticeable as the Cu content increases, which also enhances the spin-glass-like features. Further, in the x¼0.30 alloy the alignment imposed by applied magnetic fields higher than 4 kOe prevail over the configuration determined by the frustration mechanism that characterizes the spin glass-like phase.