The crystallization behaviors and magnetic properties for $Fe_{81-x}Al_{4}B_{10}Zr_{5}Cu_{x}$ (x=0, 1, 2 at%) alloys is investigated. By the addition of 1~2 Cu, the temperature range, where a single bcc phase exists, expands largely over 200 K and the grain size of bcc phase represents to less than 10 nm. For the optimally annealed Cu-added alloys, the high $mu_{e}$ (1 kHz) above 20000 combined with the high $B_{10}$ of about 1.4 T is obtained in nanocrystalline state. The low core loss of 95.8 W/kg at 0.1 T and 100 kHz is confirmed for the nanocrystalline $Fe_{80}Al_{4}B_{10}Zr_{5}Cu_{1}$ alloy.