  Motivated by future gravitational waves observations, we perform forecasts
analysis to constrain a possible time variation of the fine structure constant
($\alpha$) within the context of the so-called runaway dilaton model. For this
purpose, some gravitational-wave standard sirens mock data within the
perspective of Einstein Telescope and LISA mission were considered jointly with
current strong gravitational lensing systems observations. We find that future
standard sirens observations can also play an important role in the search for
possible variations of $\alpha$ within the methodology presented in this work.
