Conventional Fe-based soft magnetic alloy ribbons each containing Co and Ni have a problem that magnetic anisotropy that is neatly arranged in one direction cannot be induced easily even by a magnetic field annealing treatment and, therefore, a wound magnetic cores, a problem that a residual magnetic flux density Br is high, a problem that the hysteresis of the B—H curve becomes large (coercivity Hc becomes large), a problem that the change in incremental permeability relative to superimposed magnetic field becomes large, and others. In order to solve the problems, provided is an Fe-based soft magnetic alloy ribbon including a Cu-concentrated region present directly below a surface of the ribbon, and a Co-concentrated region present directly below the Cu-concentrated region. Also provided is a magnetic core including the Fe-based soft magnetic alloy ribbon.