Microstructural (crystalline volume fraction and grain size), magnetization (coercive field), and saturation magnetostriction measurements in Fe73.5-xNixSi13.5B9Nb3Cu1 alloy ribbons (x=5, 10, and 20) treated by current annealing and stress-current annealing are presented. Microstructural analysis of the treated ribbons using x-ray diffraction showed a high content of the amorphous phase in the bulk. In addition, substantial changes in the crystalline state such as grain size of the samples annealed at different conditions were observed. The alloy composition also affects greatly the grain size: increase in Ni content leads to higher values of the average grain size. The evolutions of the coercive field with the two kinds of thermal treatment were analyzed, allowing us to conclude that the addition of Ni tends to reduce the magnetic softness of the original material and that the coercivities are higher in the samples treated by stress annealing than in those treated without tensile stress. On the other hand, the saturation magnetostriction decreases with the thermal treatment, which is in agreement with the microstructural behavior (structural relaxation and nanocrystallization process), although some discrepancies are found for samples with x=5.