In this work, recently submitted to Monthly Notices, I am analysing the observed anti-correlation between the central dark matter (DM) densities of the bright Milky Way (MW) dwarf spheroidal galaxies (dSphs) and their orbital pericenter distances, which poses a potential signature of self-interacting dark matter (SIDM). I investigate such possibility by analysing the range of SIDM scattering cross section per unit mass, σ/mχ, able to explain such anti-correlation. I simulate the orbital evolution of dSphs subhaloes around the MW assuming an analytical form for the gravitational potential, adopting the proper motions from the Gaia mission and including a consistent characterization of gravitational tidal stripping. The evolution of the subhaloes density profile is modelled using the gravothermal fluid formalism, where DM particle collisions induce thermal conduction that depends on σ/mχ. I find that models of dSphs, such as Carina and Fornax, reproduce the observed central DM densities with fixed σ/mχ ranging between 30 and 50 cm^2/g, whereas other dSphs prefer larger values ranging between 70 and 100 cm^2/g. These cross sections correlate with the average collision velocity of DM particles within each subhalo’s core, so that systems modelled with large cross sections have lower collision velocities. I fit the cross section-velocity correlation with a SIDM particle model, where a DM particle of mass mχ = 0.648 ± 0.154 GeV interacts under the exchange of a light mediator of mass mφ = 0.636 ± 0.055 MeV, with the self-interactions being described by a Yukawa potential. The outcome is a cross section- velocity relation that explains the diverse DM profiles of MW dSph satellites and is consistent with observational constraints on larger scales.