In this article, we investigate the anomalous precession of planets in the nearly Newtonian gravitational regime. This limit is obtained by application of the slow motion condition to the geodesic equations without altering the geodesic deviation equations. Using a non-standard expression for the perihelion advance from the Weyl conformastatic vacuum solution as a model, we can describe the anomaly in planetary precession compared with different observational data, consisting of ephemerides of planets and the Moon (EPM2008 and EPM2011) and Fienga et al.'s planetary and lunar ephemeris (INPOP10a). As a result, using the Levenberg–Marquardt algorithm and calculating the related χ-squared statistic, we find that the anomaly is statistically irrelevant, in accordance with the INPOP10a observations. As a complement to this work, we also perform an application to the relativistic precession of giant planets using observational data calibrated with EPM2011.