We have carried out a study of the velocity dispersion of the stars in the outskirts of the globular cluster omega Cen, finding that the velocity dispersion remains constant at large radii rather than decrease monotonically. The dispersion starts to be constant for an acceleration of gravity of a=2.1+0.5-0.4x 10-8 cm s-2. A similar result is obtained reanalyzing existing data for the globular cluster M15 where the profile flattens out for a=1.7+0.7-0.5x 10-8 cm s-2. In both cases the acceleration is comparable to that at which the effect of dark matter becomes relevant in galaxies. Explanations for this result within Newtonian dynamics exist (e.g. cluster evaporation, tidal effects, presence of dark matter) but require ad hoc tuning of the relevant parameters in order to make in both clusters the dispersion profile flat starting exactly at the same acceleration. We suggest that this result, together with a similar one for Palomar 13 and the anomalous behavior of spacecrafts outside the solar system, may indicate a breakdown of Newton's law in the weak acceleration regime. Although not conclusive, these data prompt for the accurate determination of the internal dynamics of as many GCs as possible.