  The possible time variation of the fundamental constants of nature has been
an active subject of research since the large-number hypothesis was proposed by
Dirac. In this paper, we propose a new method to investigate a possible time
variation of the speed of light ($c$) along with the fine-structure constant
($\alpha$) using Strong Gravitational Lensing (SGL) and Type Ia Supernovae (SNe
Ia) observations. We assume a general approach to describe the mass
distribution of lens-type galaxies, the one in favor of the power-law index
model (PLAW). We also consider the runaway dilaton model to describe a possible
time-variation of $\alpha$. In order to explore the results deeply, we split
the SGL sample into five sub-samples according to the lens stellar velocity
dispersion and three sub-samples according to lens redshift. The results
suggest that it is reasonable to treat the systems separately, but no strong
indication of varying $c$ was found.
