  The Einstein Equivalence Principle (EEP) is one of the foundations of the
theory of General Relativity and several alternative theories of gravitation
predict violations of the EEP. Experimental constraints on this fundamental
principle of nature are therefore of paramount importance. The EEP can be split
in three sub-principles: the Universality of Free Fall (UFF), the Local Lorentz
Invariance (LLI) and the Local Position Invariance (LPI). In this paper we
propose to use stable clocks in eccentric orbits to perform a test of the
gravitational redshift, a consequence of the LPI. The best test to date was
performed with the Gravity Probe A (GP-A) experiment in 1976 with an
uncertainty of $1.4\times10^{-4}$. Our proposal considers the opportunity of
using Galileo satellites 5 and 6 to improve on the GP-A test uncertainty. We
show that considering realistic noise and systematic effects, and thanks to a
highly eccentric orbit, it is possible to improve on the GP-A limit to an
uncertainty around $(3-4)\times 10^{-5}$ after one year of integration of
Galileo 5 and 6 data.
