We have investigated the local structure of amorphous oxides in EuO-TiO2 system to understand the mechanism that gives rise to curious ferromagnetic properties recently observed in the system. X-ray absorption spectroscopy has been performed on amorphous EuTiO3 and Eu2TiO4 thin films prepared by the pulsed-laser-deposition method. The Ti K-edge X-ray absorption near-edge structure (XANES) and extended X-ray absorption fine structure (EXAFS) data analyses reveal that the coordination number of Ti4+ in the amorphous thin films is around 4, which is in sharp contrast to their crystalline counterparts, where Ti4+ ions are octahedrally coordinated by oxide ions. It is therefore inferred that Ti4+ acts as a network-forming cation in the random network structure of the amorphous oxides. From the Eu L-3-edge EXAFS data analyses, it is found that the coordination number of Eu2+ is much lower and the nearest Eu-O bond length is shorter in the amorphous thin films than in the crystalline counterparts. The amorphization-induced enhancement of ferromagnetic interactions observed in the EuO-TiO2 system is discussed in terms of the local environment around Eu2+ ions.