The perihelion advance of the orbit of Mercury has long been one of the observational cornerstones for testing General Relativity (G.R.) The main goal of this paper is to discuss how, presently, observational and theoretical constraints may challenge Einstein's theory of gravitation characterized by β=γ=1. To achieve this purpose, we will first recall the experimental constraints upon the Eddington-Robertson parameters γ,β and the observational bounds for the perihelion advance of Mercury, Δωobs. A second point will address the values given, up to now, to the solar quadrupole moment by several authors. Then, we will briefly comment why we use a recent theoretical determination of the solar quadrupole moment, J2= (2.0 ± 0.4) 10-7, which takes into account both surfacic and internal differential rotation, in order to compute the solar contribution to Mercury's perihelion advance. Further on, combining bounds on γ and J2 contributions, and taking into account the observational data range for Δωobs, we will be able to give a range of values for β. Alternatively, taking into account the observed value of Δωobs, one can deduce a dynamical estimation of J2 in the setting of G.R. This point is important as it provides a solar model independent estimation that can be confronted with other determinations of J2 based upon solar theory and solar observations (oscillation data, oblateness) Finally, a glimpse at future satellite experiments will help us to understand how stronger constraints upon the parameter space (γω J2) as well as a separation of the two contributions (from the quadrupole moment, J2, or purely relativistic, 2α2+2αγ–β) might be expected in the future.