The X-ray radiation-induced attenuation (RIA) growth kinetics are studied online in different single-mode aluminosilicate optical fibers in the near-IR (NIR) domain to evaluate their potential in terms of dosimetry. The optical fibers differ by Al contents, core sizes, drawing parameters, and also by a preform deposition process. The data show no dependence of the RIA on all these parameters, a positive result for the design of point or distributed radiation detectors exploiting RIA to monitor the dose. The RIA growth rate is unchanged for dose rates changing from 0.073 to 6.25 Gy(SiO2) s−1, and the RIA linearly increases with the dose up to 2 kGy(SiO2). Small but noticeable RIA changes are observed when the irradiation temperature increases up to 50 °C during successive irradiation runs. Such results, and the post-irradiation RIA recovery, have to be considered for the application, as they can affect the dose measurement accuracy. Finally, the spectral analysis shows no dependence of the spectral shape on the fiber and irradiation parameters. As a consequence, the data reported at 1310 and 1550 nm give information not only for the RIA kinetics at telecommunications and sensor wavelengths but also for the whole NIR range often used fiber-based technologies.