The Berry phase is a widely recognized phenomenon, extensively studied for Dirac particles interacting with an electromagnetic field. We expect that Dirac particles interacting with gravity should also cause a similar effect. We find a covariant formalism of Berry-like connection in the presence of Einstein gravity, using the WKB approximation. Using this Berry connection we can describe the Berry phase that is acquired by the Dirac particle. We perform the same analysis on massless Dirac particles (Weyl particles). Our findings indicate that, in the massless case, the Berry connection resembles that of electromagnetic waves, differing only by a proportionality factor of 1/2.