Phase-precipitation studies have been performed on samples of the metallic glasses Fe79B16Si5 and Fe78B13Si9, heated in the range 300-475 degrees C for various times (1-16 h) using Fe-57 Mossbauer transmission spectroscopy and X-ray diffraction methods. These measurements have helped in identifying the temperature ranges and annealing durations in which the amorphous structure of these metallic glasses is retained. The results revealed that the thermal stability increases as boron is replaced by silicon in the Fe-B-Si metallic glasses and that these alloys remain amorphous below 450 degrees C. The various phases precipitated above this temperature were identified as alpha-Fe, alpha-(Fe, Si), Fe3B, and Fe2B. The direction of magnetization in the two metallic glasses appears to change upon annealing.