A mostly single bcc phase with nanoscale grain sizes of 10 to 20 nm was found to form by annealing amorphous Fe-Zr-B, Fe-Hf-B, and Fe-M-B-Cu(M=Ti, Zr, Hf, Nb, and Ta) alloys for 3.6 ks in the range of 723 to 923 K. The high permeability (μe) above 10 000 at 1 kHz combined with high saturation magnetization (Bs) above 1.5 T was obtained for the bcc alloys. The highest μe and Bs values reach 14 000 and 1.7 T for Fe91Zr7B2, 20 000 and 1.55 T for Fe87Zr7B5Cu1, and 48 000 and 1.52 T for Fe86Zr7B6Cu1. Magnetostriction (λs) decreases significantly by the phase transition from amorphous to bcc phase and is measured to be 1 × 10-6 for the bcc Fe86Zr7B6Cu1 alloy. The small λs as well as the small grain size is concluded to be the reason for the good soft magnetic properties. The lattice parameter of this bcc phase is 0.2870 nm being larger than that of pure α-Fe. The small λs seems to be achieved by the dissolution of solute elements above an equilibrium solubility limit. The bcc Fe86Zr 7B6Cu1 alloy also shows the low core loss of 0.066 W/kg at 1 T and 50 Hz, which is considerably smaller than that of amorphous Fe78Si9B13 and bcc Fe-3.5mass%Si alloys in practical uses as core materials in transformer.