By means of a formal analogy with the Aharonov-Bohm effect, the Sagnac time delay and the corresponding Sagnac phase shift in the Kerr-Newman and Reissner-Nordstrom spacetimes are discussed. We find that the effect depends on the properties of the source of the gravitational field. The contributions made by the electric charge of the gravitational source can be employed to weaken it in the Kerr-Newman spacetime, even if a phase shift and a time delay still appear. This is due to the properties of the rotating source of the gravitational field.