In this work, we constraint a possible time variation of the fine structure constant (α) within the context of the so-called runaway dilaton model. The limits are performed by using the pantheon supernova Ia sample and strong gravitational lensing data, and we find that in light of the current tension on the supernova absolute magnitude MB, a varying-α can be non-null at ∼2σ confidence level. Motivated by this aspect, and within the methodology presented here, we perform a forecast analysis based on the generation of some gravitational-wave standard sirens mock data within the perspective of Einstein Telescope and LISA mission. We find that future standard sirens observations can come to play an important role in discrimination such theories.