Recently, the size reduction of magnetic components such as choke coils and linear inductors is strongly needed for the miniaturization of electronic devices. Although ferrite cores with air-gaps have been widely used for this purpose up to now, small saturation magnetization and leakage flux from air-gaps have prevented a further size reduction of magnetic components. This contribution reports that toroidal cores for choke coils with extremely low loss were developed from nanostructured FeCuNbSiB ribbons. The relative permeability of the cores was set to approximately 250 by using the creep-induced anisotropy. Deterioration of magnetic properties due to formation of a toroidal core was suppressed by controlling the core diameter and/or magnetostriction. Resultantly, the permeability of the developed cores was kept constant up to 1 MHz. The magnetic loss was much smaller than the loss values reported previously for nanostructured and amorphous cores with low permeability, and agreed with the calculated classical eddy current loss. As for DC-bias properties, the permeability and magnetic loss were kept constant up to the DC-bias field being three times higher than that applicable to a ferrite cut core with the same permeability value. These superior magnetic properties can be attributed to the large saturation magnetization and magnetization process due to rotation of magnetization as well as the nanostructure of the cores.