During overbalanced drilling, sonic measurements in a borehole are generally made in the presence of overburden, radial, and hoop stresses caused by a borehole fluid overpressure. The associated static deformations of the formation in the borehole vicinity may have both elastic and plastic strains prior to the initiation of tensile fractures. The influence of such static deformations on the borehole Stoneley and flexural dispersions has been calculated from a perturbation model for equations of motion for small dynamic fields superposed on a static bias. The static bias refers to the prestress and associated elastic and plastic strains in the formation prior to any tensile fracture. Computational results for the Castlegate sandstone formation show characteristic differences in the Stoneley and flexural dispersions, notably at higher frequencies. For a borehole of diameter 20 cm (8 inches) with an overpressurization of 10 MPa in a formation with a confining isotropic stress of 5 MPa both the Stoneley and flexural mode velocities increase by ~5--10% at frequencies larger than 5 kHz compared to model predictions that do not account for the prestress. When the overpressure is increased to 20 MPa, there is a slight increase in the plastic deformation close to the borehole surface. Most of the changes in the Stoneley and flexural dispersions in the intermediate frequency range (3--8 kHz) are caused by an increase in the borehole pressure rather than a small increase in the plastic deformation. These changes in borehole dispersions are measures of formation stress sensitivity and the radial extent of stress-induced alteration in the borehole vicinity. However, the Stoneley dispersion for frequencies up to 2 kHz and flexural dispersion for frequencies up to 3 kHz are largely insensitive to the near-field stresses caused by borehole pressures and any associated plastic deformation extending to ~5 cm into the formation in a 20 cm borehole. ¿ 1999 American Geophysical Union