The transport coefficients of strongly interacting matter are currently subject of intense theoretical and phenomenological studies due to their relevance for the characterization of the quark–gluon plasma produced in ultra-relativistic heavy-ion collisions (uRHIC). We discuss the connection between the shear viscosity to entropy density ratio, η/s, and the electric conductivity, σel. Once the relaxation time is tuned to have a minimum value of η/s=1/4π near the critical temperature Tc, one simultaneously predicts σel/T very close to recent lQCD data. More generally, we discuss why the ratio of (η/s)/(σel/T) supplies a measure of the quark to gluon scattering rates whose knowledge would allow to significantly advance in the understanding of the QGP phase. We also predict that (η/s)/(σel/T), independently on the running coupling αs(T), should increase up to about ∼20 for T→Tc, while it goes down to a nearly flat behavior around ≃4 for T≥4Tc. Therefore we in general predict a stronger T dependence of σel/T with respect to η/s that in a quasi-particle approach is constrained by lQCD thermodynamics. A conformal theory, instead, predicts a similar T dependence of η/s and σel/T.