We have synthesized Fe$_{1+y}$Te thin films by means of molecular beam epitaxy (MBE) under Te-limited growth conditions. We found that epitaxial layer-by-layer growth is possible for a wide range of excess Fe values, wider than expected from what is known on the bulk material. Using x-ray magnetic circular dichroism spectroscopy at the Fe L$_{2,3}$ and Te M$_{4,5}$ edges, we observed that films with high excess Fe contain ferromagnetic clusters while films with lower excess Fe remain nonmagnetic. Moreover, x-ray absorption spectroscopy showed that it is possible to obtain films with very similar electronic structure as that of a high quality bulk single crystal Fe$_{1.14}$Te. Our results suggest that MBE with Te-limited growth may provide an opportunity to synthesize FeTe films with smaller amounts of excess Fe as to come closer to a possible superconducting phase.
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