A theoretical model for the instability of two-dimensional turbulent boundary layer over compliant surfaces is described. The principal Reynolds stress is modelled by a well-established mixing-length eddy-viscosity formulation of van Driest. The perturbations of the mean velocity and Reynolds stress fields are coupled via the turbulence model. The investigation of instability is carried out from a time-asymptotic spatio-temporal perspective that classifies instabilities as being either convective or absolute. The occurrence of convective and absolute instabilities over viscoelastic compliant layers is elucidated. Compliant surfaces with low damping are susceptible to convective instability, which gives way to an absolute instability when the surfaces become highly damped. The theoretical results are compared against experimental observations of surface waves on elastic and viscoelastic compliant layers.