The modified Newtonian dynamics (MOND) have been formulated as a modification of the Poisson equation for the Newtonian gravitational field. This theory generically predicts a violation of the strong version of the equivalence principle and as a result, the gravitational dynamics of a system depend on the external gravitational field in which the system is embedded. This so-called external field effect has been recently shown to imply the existence of an anomalous quadrupolar correction, along the direction of the external galactic field, in the gravitational potential felt by planets in the Solar system. In this paper, we confirm the existence of this effect by a numerical integration of the MOND equation in the presence of an external field and compute the secular precession of the perihelion of planets induced by this effect. We find that the precession effect is rather large for outer gaseous planets and in the case of Saturn, it is comparable to published residuals of precession obtained by Saturn range tracking data. The effect is much smaller for inner planets, but in the case of the Earth, it appears to be in conflict for most of the MOND functions μ(y) with the very good constraint on the perihelion precession obtained from Jupiter very long baseline interferometry data. The MOND functions that are compatible with this constraint appear to have a very rapid transition from the MONDian regime to the Newtonian one.