  A new method based on large-scale structure observations is proposed to probe
a possible time variation of the fine-structure constant ($\alpha$). Our
analyses are based on time-delay of Strong Gravitational Lensing and Type Ia
Supernovae observations. By considering a class of runaway dilaton models,
where the cosmological evolution of the fine-structure constant is given by
$\frac{\Delta \alpha}{\alpha} \approx -\gamma \ln{(1+z)}$, we obtain limits on
the physical properties parameter of the model ($\gamma$) at the level
$10^{-2}$ ($1\sigma$). Although our limits are less restrictive than those
obtained by quasar spectroscopy, the approach presented here provides new
bounds on the possibility of $\frac{\Delta \alpha}{\alpha} \neq 0$ at a
different range of redshifts.
